An early scene in the Rockfort Island training facility involving a survivor in a hazard suit getting crushed against an observation window while Claire watches helplessly is an almost shot-for-shot homage to Return of the Living Dead 3. The Bandersnatches were also supposedly based off of the zombie that was in that scene.
The Bandersnatch actually bears a strong resemblance to the "Tarman" zombie from the original The Return of the Living Dead. If only he said BRAINS!
The name 'Bandersnatch' comes from Lewis Carroll's work.
Alfred acting as Alexia is just like Norman Bates. In addition, the layout of the twins' private residence seems like a further Psycho reference.