According to Wikipedia a bandersnatch is a creature that appears in a novel (Through The Looking Glass) and a poem (The Hunting of the Snark)by Lewis Carroll.
While never described in detail, the bandersnatch supposedly has ‘a long neck and snapping jaws’ and is ferocious and extraordinarily fast.
From what I can tell from the trailer the video game programmer at the center of the story appears to be taking drugs (anti-psychotic?) to stay centered, but to unlock his creativity ditches them, which causes the line between what’s real and what’s not to blur dangerously.
There’s even a scene of him literally going through the looking glass, which may be a bit on the nose.
I’m also getting a Videodrome vibe (which is always a good thing).