When Paul Thomas Anderson cast Joanna Newsom as the all-seeing mystic Sortilège in his 2014 film adaptation of the Thomas Pynchon novel Inherent Vice, he knew what he was doing. “I always think she knows a little more than the rest of us,” the director told The New York Times when he was among the interviewees for a recent profile piece on Newsom. “There is a part of her that has one foot in an alternative universe.”

Newsom’s palpable otherness has long distinguished her music, too. Variously described as a “freak folk Cinderella”, “the woman who made the harp hip”, and (less reverently) someone whose voice shares certain tics with that of Lisa Simpson, this California-born singer-songwriter inhabits a deeply literate world underpinned by a virtuosic, highly evolved musicality. Like Kate Bush and Björk before her, Newsom is that rare thing: an utterly unique species whose kookiness isn’t a construct, but rather something in the genes

Divers is her fourth album, and like its triple LP predecessor, 2010s Have One on Me, it is not for the faint-hearted. Part of the accompanying press-release describes it as “a family of polysemic song-sets; a … Liederkreis of harmonic sympathies and knotted hierarchies”, and there is plenty more in that vein.

Reading the full thrust of this highfalutin fanfare, part of you feels inclined to consign Divers to the deep, but to do so would be a grave error. A profoundly enigmatic record with few easy handholds Newsom’s latest may be, but boy does it reward patience. Divers is a Russian doll of an LP, the perfect work for those who like their albums to reveal new treasures with each listen.

Beginning with a soundscape of birdsong, distant drums and other, less easily identifiable elements, opening song Anecdotes ratchets up Newsom’s by-now-familiar daring. Quickly moving to piano, vocals and icicle-crisp harp, it grows, over the course of six-and-a-half minutes, to take in clarinet, English horn, Baroque flourishes, complex modulations, ragtime elements, strings and more.

The song’s densely poetic lyrics might conceivably view warfare through the eyes of a nightjar (various bird species crop up on Divers), but you certainly wouldn’t bet on it. What’s intriguing, though, is that Newsom has stressed that the countless layers of meaning and allusion she has so artfully woven into Divers can be decoded into a coherent, structured narrative.

Where the album’s playful, fiendishly intricate first single Sapokanikan is concerned, your scribe has thus far only gleaned that its title name-checks the Native American settlement upon which Manhattan’s Greenwich Village now stands, and that Newsom alludes to Ozymandias, Shelley’s 1818 sonnet. The most tenacious of Newsom’s famously obsessive fans will doubtless be ahead of me with the decoding process, feverishly blogging their own hunches.

The nearly six-year gap between Have One on Me and Divers must have concerned Newsom’s record label, but the 33-year-old singer is now bankable enough to dictate her own timetable. Working with a gifted coterie of arrangers including Nico Mulhy, the classical composer and sometime collaborator of Sufjan Stevens and Björk, Newsom spent the lion’s share of the last two years perfecting her new, self-produced album’s meticulous overdubs. But before that her fans clocked a distinct lack of urgency; a woman who was simply living life, having fun.

In 2011 Newsom sang for The Muppets film. In 2012 she sent herself up on the hit US television comedy Portlandia. Then, in September 2013, Newsom married US actor and comedian Andy Samberg. The couple have since taken-up residence in a Los Angeles mansion that Charlie Chaplin rented in the 1920s.

Early days, perhaps, but domesticity, so often cited as creativity’s nemesis, certainly hasn’t blunted Newsom’s thrust. Indeed, the title track of Divers, another slow-evolving epic, finds her at the peak of her powers, her harp more handloom than instrument as she weaves transporting arpeggios and sings: “And in an infinite capsize / Like a bull tearing down the coast /double hulls bearing double masts / I don’t know if you loved me most / but you loved me last.”

With its magical, trad-Scottish coda, Waltz of the 101st Lightborne is similarly dazzling, while the album’s lone cover-version, Same Old Man, is a fresh, hugely evocative take on that traditional folk song, spare banjo pecks carefully aligned with Newsom’s stacked vocal harmonies and woozy Mini-Moog synthesiser.

Despite the aforementioned opacity of most of the lyrics on Divers, certain overarching themes eventually emerge. “The longer you live, the higher the rent,” sings Newsom on Leaving the City, and the passage of time and the past’s ever-emotive relationship with the present figure elsewhere, too. Goose Eggs, with its warm-fuzz Fender Rhodes piano and electric harpsichord, intrigues from the outset: “What we built / at the kiln that won’t be stilled / did not set well.”

Given that Newsom has said that the 10 originals on Divers are, at root, love songs, there might be good reason for all this beautiful obfuscation. One can also understand why a woman who is part of a celebrity couple might covet a carapace of metaphor and allusion.

Newsom has actually gone further than that, though, stating that she is not this album’s narrator. We must bear that in mind listening to The Things I Say, a succinct and – save for its use of musical saw – fairly conventional confessional at the piano. “There’s an old trick played / when the light and the wine conspire / to make me think I’m fine”, sings the song’s protagonist. “I’m not, but I have got half a mind / to maybe get there, yet.”

Divers is a startling feat of the imagination, a record that lifts Joanna Newsom head and shoulders above much of the competition. Few writers are able to combine such mastery of grand themes with such a singular, compelling voice.

James McNair also writes for Mojo magazine and The Independent.