Haruki Murakami has always been a cult writer, if one can say that about a novelist who regularly sells millions, both in his native Japan and in translation. Well, 1Q84 – an epic romance in three "books" and two volumes (Book 3, translated by Philip Gabriel, is published separately) – is his cult novel. In Underground (2000), Murakami interviewed former members of the Aum sect and survivors of its 1995 nerve-gas attack on the Tokyo subway. In that book, he implicitly promised a fictional engagement with the subject of cults; now he has delivered.

At least two cults are active in this story. One is a Christian sect known as the Society of Witnesses, whose pamphleteering members refuse lifesaving surgery. The second cult is more Aum-ish and more mysterious. It is called Sakigake (which might mean "forerunner", "precursor", or "pioneer"), and from two wounded escapees we hear some very nasty things about its leader.

Other groupings in the novel can also seem cult-like in structure. One of the two main characters is a maths teacher and writer, Tengo, who gets drawn by his editor into a literary conspiracy: he ghost-rewrites a novel by a teenage girl, which then wins a prize and becomes a bestseller. (Murakami's translators Rubin and Gabriel, assigned a volume each to meet a rush publishing schedule, have also conspired successfully in producing an English version of limpid consistency.) Elsewhere in Tokyo, an elderly woman known only as the Dowager runs a shelter for female victims of domestic violence. To inflict clandestine punishment on the brutish men, the Dowager retains the services of the novel's other main character: a woman named Aomame, a martial-arts instructor and physical therapist.

Echoes, here, of the themes of Stieg Larsson's Millennium trilogy, but what doesn't often happen in Swedish noir is that groups of terrifying leprechauns emerge from the mouths of goats and people, or that characters look up and see a second moon, misshapen and moss-coloured, hanging in the sky. Yes, this is a Haruki Murakami novel, where magical and dreamlike phenomena are deadpanned into existence with the same calm craft that his characters routinely employ in cooking themselves delicious-sounding Japanese meals:

He put the sliced celery and mushrooms into the frying pan. Turning the gas flame up to high and lightly jogging the pan, he carefully stirred the contents with a bamboo spatula, adding a sprinkle of salt and pepper. When the vegetables were just beginning to cook, he tossed the drained shrimp into the pan. After adding another dose of salt and pepper to the whole thing, he poured in a small glass of sake. Then a dash of soy sauce and finally a scattering of Chinese parsley. [...] He took a fresh beer from the refrigerator, sat at the kitchen table, and, still lost in thought, proceeded to eat the steaming food.

Nearly all Murakami's novels play with the device of a parallel dimension into which characters can slip through cracks or portals, such as (here) an emergency staircase leading down from a city expressway. The novel is set in 1984, but when Aomame sees a news report about the construction of a joint American-Soviet moon base, and then a second moon in the sky, she deduces that she has stumbled into a different universe, which she christens 1Q84: the "Q" stands for "question". Alternate worlds, in previous Murakami works such as The Wind-Up Bird Chronicle and Sputnik Sweetheart, have usually been places where a man is looking for a woman he has lost. The same is true here, except that the search is mutual, and 1Q84 worries more disconcertingly at the possibility of becoming "irretrievably lost", a phrase that appears several times, growing ever creepier. Once you have crossed over, there is no guarantee of successful navigation or escape: "She sensed that she had found her way into a region that was completely foreign to her – a deep ocean trench, say, or the surface of an unknown asteroid: the kind of place it might be possible to reach with great effort, but from which return was impossible."

Even in Murakami's Norwegian Wood, a purely naturalistic novel, the hero speaks of being "tossed" into a "labyrinth" to describe his impossible situation. In 1Q84, meanwhile, Tengo is lured into "chaotic territory" and calls the (real) world "labyrinth-like", while Aomame's own brain itself becomes a maze: "her cerebrum – the gray labyrinth where consciousness resided".

In outline, this vast novel's plot is elementary: boy and girl meet, part, and look for each other, with the kind of melancholy yearning that Murakami has long had tuned to a high art. The novelist has said, however, that he wanted to make this "simple" story as "complicated" as possible. That he has certainly accomplished, and the book's sheer length virtually guarantees that a certain scene near the end, in a playground, will be tremendously affecting. In order to make sure, however, Murakami has had the courtesy to write it with exquisite tact. It is a scene in which complete mastery of technique makes technique vanish: as perfect as any two pages might hope to be.

Disparate pleasures are each given time to grow rich by the novel's long span, especially a sad, funny and ultimately wrenching portrait of a private detective, who unexpectedly becomes a third focus of third-person narration in the third book. 1Q84 is not self-consciously hard-boiled in the manner of Murakami's early fiction (say, A Wild Sheep Chase), but it builds up gradually to a few passages of extremely potent suspense. There are also some rumbustious sex scenes (treated with the same tolerant, wholesome curiosity that Murakami devotes to the stir-frying of food), as well as a cluster of details that function as generous and reassuring fan-service, such as a young woman with a beautiful ear, or the obligatory apparitions of cats and crows.

Uncanny mysteries haunt the novel: a nurse who claims to be a reincarnation (but of whom?), and a peculiarly insistent TV licence-fee collector who is heard but never seen. Little voices irrupting into the text seemingly from nowhere ("'Ho ho,' says the keeper of the beat"), meanwhile, begin to evoke the slow-burning dread of Don't Look Now and Rosemary's Baby. Near-despair is rarely too far, though, from a certain forlorn comedy. At one doleful turn, "Tengo started to have doubts about the difference between a person being alive and being dead. Maybe there really wasn't much of a difference to begin with, he thought. Maybe we just decided, for convenience's sake, to insist on a difference." At length, however, Tengo decides that there is a difference after all: "Being alive, if you had to define it, meant emitting a variety of smells." (Murakami has said that his favourite film director is Aki Kaurismäki, the hilariously dismal Finn. That fits.) The slipped cosmos of 1Q84 encompasses, as well, narratives of even stranger worlds: in particular, a superlatively eerie fable about a "Town of Cats", which even devoted Murakami admirers would not want to visit.

Some have considered Murakami's deployment of fantastical elements in his fiction to be fey or under-justified. His own reasoning about the practice, in a 2004 Paris Review interview with John Wray, is revealing: "We are living in a fake world; we are watching fake evening news. We are fighting a fake war. Our government is fake. But we find reality in this fake world. So our stories are the same; we are walking through fake scenes, but ourselves, as we walk through these scenes, are real. The situation is real, in the sense that it's a commitment, it's a true relationship." So, too, in Murakami's novels, events might be unnatural and outré, but the characters are as human as possible. Murakami achieves this in two ways: first, by an unrushed, tender cataloguing of small daily action (preparing "steaming food"), and second, by the lovingly humorous imagining of his characters' inner chatter. Here is Aomame, in a moment of downtime: "That was the most she could get herself to do – stare at the ceiling. Not that the ceiling had anything of interest about it. But she couldn't complain. Ceilings weren't put on rooms to amuse people."

Cultural touchstones help to anchor people in Murakami's shifting realities. There are references in 1Q84 to Chekhov, Stanley Kubrick, Dostoevsky, Lewis Carroll, Macbeth, and Carl Jung, though the most important works cited are musical. Janácek's Sinfonietta plays a major role; an investigator muses playfully on Sibelius; and a woman discourses in bed on her love for the jazz clarinettist Barney Bigard. Murakami ran a jazz bar before becoming a writer, and music – jazz, classical, or very occasionally pop – always leaks into his prose. But the special importance of music in this novel is a key to its major theme, which is time: the suspicion that time is an illusion, the yearning to recapture the past and the experience of how time can get "deformed", knotting itself "like a tangled string". In this respect the novel's explicit (and usually creaking) references to Orwell's Nineteen Eighty-Four (the cult as totalitarian state, "making mindless robots"; the cult leader – or God himself – as "Big Brother") are something of a red herring; Murakami's deep concerns here are more Proustian, which he signals when he has Aomame, holed up in hiding, read (or try to) In Search of Lost Time.

"Surely," thinks Tengo, for his part, "if a person had to accept the time through which he had just passed uniformly in the given order, his nerves could not bear the strain. Such a life […] would be sheer torture." Music redeems time by patterning it with meaning, an idea suggested when we learn elsewhere of Tengo's pleasure in playing the timpani at school: "He felt a natural joy in dividing time into small fragments, reassembling them, and transforming them into an effective row of tones." Much later, this picture finds its rhyme in a beautiful vision of mutability: "A ragged line of sparrows sat on an electrical line, constantly switching positions like musical notes being rewritten." (In Murakami, a bird is rarely just a bird.) The characters fantasise about forgetting time altogether, but such grace is afforded here only to the ghosts of metaphor: "A weather front was stalled out in the Pacific – like a lonely person, lost in thought, oblivious of time."

Writing in the New York Times last year, Murakami called himself "a teller of stories […] a hopefully humble pilot of the mind and spirit". His diagnosis of cults, in both Underground and 1Q84, is that they are dangerously effective feeders of people's need for stories. As Sakigake's leader, the Kurtz figure at this novel's crepuscular heart, says: "Almost no one is looking for painful truths. What people need is beautiful, comforting stories that make them feel as if their lives have some meaning."

Do Murakami's stories themselves make people feel as if their lives have some meaning? Some critics are unsure what to make of him, the prejudice being that a writer who is so popular, particularly among young people, cannot really be that good, even if he is now quoted at short odds each year to win the Nobel prize for literature. But Murakami's success speaks to a hunger for what he is doing that is unusual. Most characters in the modern commercial genre called "literary fiction" take for granted a certain unexamined metaphysics and worry exclusively about the higher-level complexities of circumstance and relationships. Throughout Murakami's oeuvre, on the other hand, his characters never cease to express their bafflement about the nature of time, or change, or consciousness, or moral choice, or the simple fact of finding themselves alive, in this world or another. In this sense, Murakami's heroes and heroines are all philosophers. It is natural, then, that his work should enchant younger readers, to whom the problems of being are still fresh, as well as others who never grew out of such puzzlements – that his books should seem an outstretched hand of sympathy to anyone who feels that they too have been tossed, without their permission, into a labyrinth.