Just when you thought Nicolas Cage’s filmography couldn’t get any weirder, along comes Kristoffer Borgli’s “Dream Scenario” to mess with your head.

Cage plays a character you probably wouldn’t notice in real life: Paul Matthews. Schlubby, balding, in rumpled pants and brown leather loafers, he’s a tenured professor at a university you’ve never heard of, droning on year after year about collective consciousness and the wisdom of the herd. And then something weird happens. Paul starts to appear in people’s dreams, either standing around or just strolling through, and suddenly this all-but-invisible man has people paying attention to him. What does Paul do? What would you do in those terrible shoes of his?

A Norwegian helmer making his English-language debut for A24 (by way of Ari Aster’s production company), “Sick of Myself” director Borgli takes this surrealist premise and approaches it in with a tone that’s a singular mix of comedic and sinister. Is this a fantasy? A fable? A new kind of horror movie? Actually, “Dream Scenario” is all of the above and then some, for it also shares a certain postmodern DNA with two of Cage’s most boundary-pushing movies, “Adaptation” and “The Unbearable Weight of Massive Talent.” Paul Matthews may be a Paul Giamatti type, but it’s not just some random college professor popping up in people’s heads unbidden. It’s Nicolas “not the bees!” Cage, an actor so distinctive, he may well have made bizarre cameos in your dreams already.

Here, as in “Adaptation,” Cage has the challenge of trying to look unremarkable. Costume designer Natalie Bronfman helps, dressing the star in drab clothes the character probably ordered from a catalog. So does the hair and makeup team, who’ve shaved his trademark widow’s peak and let the gray show through in his beard. In class, Paul lectures about zebra stripes, explaining how their camouflage works: The idea isn’t to blend in with nature, but to disappear into the crowd. That’s been Paul’s strategy most his life, though he does have the desire to publish. He’s been working on a theory about ants (something related to their superorganism status, it sounds like) and underhandedly confronts a former classmate (Paula Boudreau) about a paper overlapping with research he did decades earlier in grad school.

In short, Paul is a pathetic character whose early behavior speaks volumes about both his ambitions and his incapacity to follow through on them. He has two daughters (Lily Bird and Jessica Clement), to whom he is hopelessly uncool, and a wife, Janet (Julianne Nicholson), who supports him. Then the dream thing starts to happen, disrupting his relatively stable home life. The movie opens with a mild nightmare, which seems normal enough until impossible things start to happen. Borgli shoots the dreams no different from the waking moments, creating a kind of mind game for audiences, who must determine at any given moment whether the scene in question is real or imagined. Or maybe it’s all just a dream.

The movie’s point of identification here is Paul, so it’s slightly unusual (but also the most entertaining thing about the movie) that “Dream Scenario” allows us into the subconscious of other characters. Paul may appear in their dreams, but he’s not the one making it happen, and he has no access to those memories. Still, once the world realizes that this seemingly unremarkable everyman is appearing in nearly all of their heads, Paul becomes a kind of overnight celebrity. His sparsely attended classes quickly hit capacity, and he fields questions from strangers who share their dreams. There’s the young woman who pictured Paul while she was being attacked by crocodiles, and the guy who saw Paul walk by as he was pulling a tooth from his mouth.

No explanation is ever given for why Paul is suddenly “invading” strangers’ dreams. Nor does the film answer why certain people aren’t affected (including his own wife). As an evolutionary biologist, Paul seems better equipped to solve this mystery than most, and audiences — like some in the film — may wonder if he’s secretly doing something to cause the phenomenon. It’s just one of those things, like the door that leads to a passage through the interior of John Malkovich’s head in “Being John Malkovich,” that is what it is because the movie says it’s so. The point, as in that Charlie Kaufman movie, is how the characters react to something so extraordinary and what that reveals about them.

Paul embraces the situation at first, agreeing to a handful of TV interviews. He even flies to New York, where he takes a meeting with an unconventional marketing firm called “Thoughts?” In a long and uncomfortable but very funny pitch session, Trent (Michael Cera, playing a shaggy Harmony Korine type) sits at the other end of a conference room and tries to convince Paul — whom he calls “the most interesting man in the world right now” — to become world’s first dream influencer. In this and a dozen other ways, the timeless-looking film engages with up-to-the-minute ideas, from cancel culture to going viral, ultimately revealing itself to be a sly social satire.

Still, Borgli keeps us guessing, especially as the way Paul appears in people’s dreams starts to change. A young woman (Dylan Gelula) who works for Thoughts? admits to having “intense sex dreams,” even going so far as to invite Paul home to reenact one together, resulting in one of the most emotionally vulnerable scenes Cage has ever performed. Others report that Paul has turned violent in their dreams, and just like that, his celebrity turns toxic. Strangers pick fights in public, someone spray-paints the word “LOSER” on his car, and the dean (Tim Meadows) is obliged to put him on leave.

Toward the end, Borgli tips his hand by introducing a technology that can project influencers into people’s dreams. The joke doesn’t explain what’s been happening so much as why, on a meta-level, Borgli might be interested in the idea: Just as dreams can be interpreted, so too can “Dream Scenario,” and this development suggests the whole film could be his take on social media — the way viral celebrities and memes invade the minds of millions of people at once, amusing us for a time, only to be rejected and potentially even reviled when their 15 minutes are up. In an uncommonly low-key performance, Cage humanizes that experience and gives us much to think about.