Norman has a third personality.
It is clear from the film that Norman believes his mother to be the murderer. However, according to "mother's" ending monologue, she can't move. So, the real murderer must be a third personality.- The third personality could be his mother before she became immobile.
- This is actually true in the book. For the book, there was Norman (who was a representation of himself as a child), Norma (his mother) and "Normal" (who was Norman as an adult).
Norman was possessed.
Hitchcock often implied ambiguous messages that much of his thriller and horror elements borderline on the supernatural. So odds are "Mother" really did kill through Norman. This is hinted at the end of Psycho IV. Norman's "mother" was the same entity as The Dark Passenger.
In line with the above WMG that states Norman was possessed. If he was, then this was the entity possessing him.