The thing is, a time travel movie is pretty much always surreal.
Even if our protagonist never comes face to face with a white-haired doppelganger, the causality of timeline mechanics is so complicated and interrelated that a clean trip will never happen. Any time travel causes instant entropy in a narrative, a problem that generally only increases as the plot thickens—and in movies, it's one that aligns more closely with the surreal than with any of science fiction's more linear cinema tropes. (Somewhere in the forest, a butterfly flaps its wings; a thousand miles away, a Bosch painting breaks out ten years ago.)
Surrealism, formally codified as a movement in the early twentieth century, rejected the real in favor of striking juxtapositions that challenged audiences to accept and try to understand the bizarre, the uncanny, and the uncomfortable. It was a natural aesthetic for the emerging medium of cinema, and aside from the obvious imagery of the surreal (1915's Alice in Wonderland featured a nightmare parade of monarchs with swollen heads and an oddly faithful game of croquet), movies quickly discovered quicker, subtler uses—the jump cut is a vestige of this technique, juxtaposing two vastly different frames and expecting audiences to connect the narrative continuation they represent, or to be affected by the emotions suggested by this contrast in an ever-growing visual vocabulary.
And cinematically, the vocabulary used to illustrate both time travel and the surreal is markedly similar; surreality is a visual cue that something has ceased to make sense, which on its deepest level is the same modus operandi of any time-travel endeavor, making the visual markers a narrative tool. Sitting across the table from someone who knows the impossible, whether it's you or a stranger, a diner on Mulholland Drive or a diner in Kansas, is always meant to seem wrong. The aesthetic elements of the surreal—juxtapositions, implausibility, unease—are the concrete supplement to the more introspective narrative elements around which time travel is most often centered: identity, choice, and struggle against inexorable force.
Terry Gilliam in particular is no stranger to the surreal—it's the milieu in which he's most at home—and there's nothing he loves more than someone doomed by an inexorable force. (Brazil was, at heart, a time-travel cautionary tale in which Jonathan Pryce's nightmare future was presented to the audience-as-protagonist.) And as such, it's no surprise that when Gilliam tackled time travel, in 12 Monkeys, he embraced the surreal in his own way. The film is largely experienced through the eyes of James Cole (Bruce Willis, whose chiseled yet comforting face must have "Send Me Back in Time, People Will Believe Me" written all over it).
There's no shortage of illogical and uneasy imagery, as each of Cole's successive returns to the past gets him branded as insane, or a fugitive, or a visionary (including the surreal within the surreal, when his doctor sees a World War I photo of him, proving his claim). Whenever he returns from the future, he encounters in small jumps a world in which his prior visits have affected others (and himself) so deeply that they seem at first to be alternate timelines. (There's a certain surreality just seeing the progression of Brad Pitt from fellow inmate to enviro-guerilla mastermind.) But because it's a Gilliam, we know the inexorable force will win, bringing home the surrealism of a man who dies in front of his own eyes.
Death as the only way out of a time loop is common in the genre, but the scope of the surreal depends on the crux of the narrative. In 12 Monkeys, the imagery provides the unsettling foreshadowing of the futility of Cole's quest to fix a world inexorably headed for disaster. But sometimes the surreal comes closer to home, and highlights the psychological ramifications of knowing the worst is heading right for you, and you've become your own worst enemy.
The poster boy for this particular trauma is Donnie Darko. The seven-foot, skull-faced man-rabbit Frank appears to Donnie late one night and warns him of an oncoming apocalypse, with Donnie absent from the site of a crash that would have killed him. After the near miss, Frank walks Donnie through the alternate timeline he's created, a monstrous motif against the hyperrealism of suburban life (though only nominally more terrifying than Sparkle Motion). Frank's knowledge of the future becomes even more unsettling and unreliable as he starts to incite the desperate Donnie to violence in an attempt to stave off the disaster. As the inevitable entropy kicks in and Donnie's loved ones are harmed, directly or sidelong, Donnie realizes Frank is a timeline-slipping mirror of himself, and that this apocalypse is personal. His blissful suicide, to cut the causality short, is one of the film's less disturbing instances of the surreal.
However, it doesn't all have to be razor-mouthed rabbits and nihilism—some films use the juxtapositions of the surreal in their time-travel causality for comedic purposes. Marty McFly uses the terrifying causality-meter of a vanishing family photo (and the equally terrifying preppiness of the Enchantment Under the Sea dance) to his advantage. It's treated with utilitarian plotcakes, used as a gauge of the success of his machinations and the remaining time left before his personal Zero Hour; there's relatively little time spent dwelling on the visceral horror of a snapshot in which your mistakes erase whole members of your family. (Perhaps that's for the best; he had a dance to get to.)
Cinema's other most successful comedic journey through time belongs to a pair of young fellows who bent the rules of a universal continuum for their history test.
In Bill and Ted's Excellent Adventure, our heroes and their far-future babysitter approach the surreal with their particular SoCal je ne sais quoi. With multiple opportunities to loop back and correct mistakes, and being lucky enough to land at moments in history when no one was taking notes, Bill and Ted are able to sail through the visual tension with the greatest of ease—traversing ancient Greece in board shorts, stashing Beethoven and Genghis Khan in a physics-bending phone booth, and dropping Joan of Arc off at the mall to lead aerobics. While played for laughs, the surreal imagery has all the hallmarks of the genre, and even includes doubling—though, given that the pair aren't overburdened with introspection, there's significantly less existential angst here than with most people realizing they're responsible for a recursive time loop on which swathes of history rest and still can't manage to wind their watches. (And of course, there's always the implied surreal within the mundane, when you consider Bill and Ted had to take several hours after the final exam to jump back in time and run causality errands.)
There are, of course, other ways for time travel to interact with the surreal: Memento's time-jumps present a surreal timeline with all the cinematic tension of time travel without the doppelganger; and even without doubling, the Terminator franchise manages to give us all the visual unease of an unstoppable robot assassin coming back in time as Surrogate Dad of the Year. But even when the general aesthetic is realistic, the easiest way for a movie to signal time travel is for something small to come apart in-frame—an older version of the protagonist, a photograph of someone at an impossible place and time—and let that uneasy space grow wider. It visually raises the stakes, and reminds you that time is a force to be reckoned with, the odds are against you. (In fact, maybe just sit back with a Gilliam movie and abandon all hope.)