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.
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.