While the rest of his cohort have fallen by the wayside or been absorbed into the Hollywood system, the film-maker has stayed weird, as his new movie of erudite vampire love reveals

The word "hipster" invariably crops up in discussions about American film-maker Jim Jarmusch, not least because he looks the part. He is tall, lean, often wears shades and has a famous shock of hair that started turning silvery grey in his teens; his basso drawl completes the uncanny resemblance to a certain Hollywood great, which inspired Jarmusch to found a jokey secret society, The Sons of Lee Marvin.

Jarmusch is without a doubt the most rock'n'roll of film-makers – although he obliges you to define the term. He has worked with a lot of musicians, either as composers or as actors – Neil Young, Tom Waits, Screamin' Jay Hawkins, hip-hop producer RZA. But if you look at the breadth of Jarmusch's references, and the enthusiasm with which he showcases them on screen – William Blake, native American mythology, lute music, Tesla's physics – then it all becomes rock'n'roll after a fashion, all glamorous grist to his intensely idiosyncratic cinema.

Jarmusch's new film, Only Lovers Left Alive, offers his most eclectic brew of such allusions. It's a vampire romance about a centuries-old couple (played by Tilda Swinton and Tom Hiddleston) who have had ample time to accrue all the erudition in the world, from the music of 17th-century English composer William Lawes to the different timbers that a wooden bullet might be fashioned from.

What fascinates Jarmusch in the vampire myth is less the usual blood-guzzling, though there's plenty of that, than the educational opportunities afforded by supernaturally extended life.

The couple might come across as a little superior in their knowingness, but Jarmusch has said: "If you'd lived for a thousand years and saw humanity devolving, you might feel a little snobbish as well." The result is a singular dark comedy that will charm some and raise the hackles of others – the Guardian's Peter Bradshaw, reviewing it in Cannes last May, found its name-dropping arch-romanticism "studenty". But Jarmusch's unabashed literacy and curiosity about the world are rare in contemporary American cinema, and marks of a defiantly individual worldview.

Jarmusch has been writing and directing for more than 30 years – starting in 1980 with his mini-feature Permanent Vacation (certainly distinctive, if today barely watchable), followed by a much surer feature debut, the wry bohemian road movie, Stranger Than Paradise (1984).

Of his generation of US independents, Jarmusch has stayed the course, and stayed weird, while others fell by the wayside (Hal Hartley) or learned to work with the mainstream (Spike Lee, the Coens). Jarmusch has sometimes worked with Hollywood names (Johnny Depp in Dead Man, Cate Blanchett in the shorts anthology Coffee and Cigarettes) but has generally stuck to modest budgets. He also reaps modest rewards: the relatively conventional Broken Flowers (2005), starring Bill Murray, grossed nearly $14m in the US, but that exceeded the American box-office total of his other eight features combined (not including Only Lovers).

His films consistently flout the conventions of American screen storytelling. For one thing, their subjects are not always primarily American, and Jarmusch often shows the US from the perspective of foreign visitors: Italian, Hungarian, Japanese. He likes to blend American and world cinema traditions: Ghost Dog: The Way of the Samurai (1999) is a hip-hop gangster movie infused with Japanese warrior lore and filtered through the French existential thrillers of Jean-Pierre Melville.

In Only Lovers, Tom Hiddleston's character has a wall of portraits apparently representing Jarmusch's own pantheon of heroes: among them, Mark Twain, Buster Keaton, Thelonious Monk, Joe Strummer.

The director's seriousness is often underestimated, says New York critic and festival director Kent Jones: "There's been an overemphasis on the hipness factor – and a lack of emphasis on his incredible attachment to the idea of celebrating poetry and culture. You can complain about the preciousness of a lot of his movies, [but] they are unapologetically standing up for poetry. [His attitude is] 'if you want to call me an elitist, go ahead, I don't care'."

Another thing that makes Jarmusch distinctive is his genuine independence: he is extremely rare in that he has made a policy of keeping control of his own negatives. But his refusal to play the industry game has not made things easy for him. When Harvey Weinstein pressed Jarmusch to cut his 1995 western Dead Man, the director stuck to his guns – later claiming that his refusal had resulted in the film being half-heartedly promoted on release.

Jim Jarmusch, Tilda Swinton and Tom Hiddleston at the premiere of Only Lovers Left Alive at the Cannes Film Festival on 25 May 2013. Photograph: John P. De Graeve/Rex Features

Jarmusch's unique sensibility doesn't always appeal to the market. It took seven years to finance Only Lovers Left Alive, with the film finding no takers in the US. In the end, the project was adopted by European producers, Reinhard Brundig in Germany and British veteran Jeremy Thomas. Thomas sees individualists like Jarmusch as an endangered species. "He's one of the great American independent film-makers – he's the last of the line. People are not coming through like that any more," he said.

Jarmusch was raised in Akron, Ohio, which is partly why his new film muses on the decline of Detroit; he grew up seeing the nearby city as "a mystical, magical place – the Paris of the midwest". In the early 70s, he briefly studied journalism, then moved to Columbia, New York, to study literature, absorbing the influences of poets such as John Ashbery and Frank O'Hara. As a student he spent 10 months in Paris, soaking up screen history at the Cinémathèque, then studied film at New York's Tisch School of the Arts, where he met his lifelong partner, Sara Driver, herself a notable independent director. He also worked as assistant to the Hollywood maverick auteur Nicholas Ray – another prominent face on the wall in Only Lovers.

Then there's the musical side of Jarmusch's career. In the late 70s and early 80s, he was a face in New York's downtown underground, singing and playing keyboards in post-punk band the Del-Byzanteens; two musicians from that scene, saxophonist John Lurie and Richard Edson, one-time drummer in Sonic Youth, starred in Stranger Than Paradise. In his soundtrack choices, Jarmusch is deeply eclectic: Coffee and Cigarettes includes Iggy Pop, the Modern Jazz Quartet and a Mahler Lied performed by Janet Baker, while Broken Flowers was immensely influential in popularising Ethiopian jazz. In this respect, Jarmusch is on a mission. "One thing about commercial films," he once remarked, "is – doesn't the music almost always really suck?"

Today, the director plays in rock trio SQÜRL and collaborates with Dutch lute player Jozef van Wissem – the two fields of activity combining magically in the eerie Only Lovers score. Van Wissem says: "I think Jim likes to be creative in a way that involves just a few people, as opposed to directing. He likes the gang idea of being in a band." All the new film's cultural references, Van Wissem says, are authentically close to Jarmusch's heart. "It's a very personal film, maybe even autobiographical. Jim is a cultural sponge, he absorbs everything."

Jarmusch's critical stock has fluctuated – Dead Man was considered a bore by some, hailed as a modern classic by others, while the glacial meta-thriller The Limits of Control in 2009 mystified quite a few diehard admirers. Meanwhile, for all his erudite thoughtfulness, the director has claimed that, for him, film-making is less about intellect than instinct. "I feel like I have to listen to the film and let it tell me what it wants. Sometimes it mumbles and it isn't very clear," he said.

Jarmusch can't be easily pinned down to any cinematic wave or category. "I don't know where I fit in. I don't feel tied to my time."

He is certainly not on the same time scheme as the rest of cinema, or indeed, the rest of humanity – which is perhaps why Only Lovers Left Alive is one of several of his films, including Night on Earth and Mystery Train, to take place after dark.

Tilda Swinton has said: "Jim is pretty much nocturnal, so the nightscape is pretty much his palette. There's something about things glowing in the darkness that feels to me really Jim Jarmusch. He's a rock star."