It's also worth pointing out that Schnitzler was a contemporary and fellow-traveler of Freud's, as images of the Freudian uncanny pop up everywhere. There's the automaton-like women in the mansion. There's the red-robed masked man with black holes for eyes. And there are doubles galore: Nuala and her friend, the two Japanese customers of Mr. Milich's (themselves doubling genders with their transvestitism), the alliterative names of the two men who lead Bill and Alice into trouble (Nick Nightingale and Sandor Szavost respectively), the masks and the faces beneath them, the two notes of the ominous Ligeti music. Even the daughter of Dr. Bill's dead patient and her husband serve as a sort of tragicomic, less attractive doubling of Dr. Bill and Alice themselves (note the placement of both the bereaved daughter and Alice in front of blue rooms, their similar hair color and style, etc.).
The final bit of doubling is another source of great vexation for the film's detractors: the repetitive dialogue. Time and time again, Bill will repeat a line just spoken to him by another character. "Maybe had Kubrick lived he might have spotted this in editing," they say--oh yeah, I'm sure he had no idea that was going on. What was he trying to achieve with this effect? Repetition is doubling, and it's also an instance of the Freudian uncanny unto itself, calling to mind non-human processes of cognition and communication (cf. the dialogue of the "twins" in The Shining). It also yields a certain narcotic, mind-altering rhythm after a time, connoting inward-facing obsessiveness and detachment from reality (cf. the "I will destroy him!" scene in Barton Fink). But there's a simpler reason, too: Bill needs things repeated to them because he simply does not understand anymore. His customary method of looking at the world has been rendered nonsensical, irrelevant, not even by deeds but by mere words. So he struggles to find a new way to frame things. He needs to repeat the new words to help make them real, to clarify them, to open his eyes to the new reality he's trying to explore. And when he does have them opened, what he sees is horrifying. That's the dream, and then that's the nightmare.