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.
Norman committed the murders, his mother was innocent.Not just physically, I mean the personalities. In some twisted way Norman, who saw his mother as a separate entity, was trying to frame her. The ending monologue could very well be seen as supporting this.
Norman's "mother" was the same entity as The Dark PassengerIn line with the above WMG that states Norman was possessed. If he was, then this was the entity possessing him.