Psycho

Marion Crane steals $400,000 from her employer to get her boyfriend, Sam Loomis, out of debt. She flees Phoenix, Arizona, by car. While en route to Sam's California home, she parks along the road to sleep. A highway patrol officer awakens her and, suspicious of her agitated state, begins to follow her. When she trades her car for another one at a dealership, he notes the new vehicle's details. Marion returns to the road but, rather than drive in a heavy storm, decides to spend the night at the Bates Motel.
Owner Norman Bates tells Marion he rarely has customers because of a new interstate highway nearby and mentions he lives with his mother Norma in the house overlooking the motel. He invites Marion to have supper with him. She overhears Norman arguing with his mother about letting Marion in the house; and, during the meal, she angers him by suggesting he institutionalize his mother. He admits he would like to do so, but he does not want to abandon her. Later that night, while Marion is changing, Norman secretly watches her from a peephole in his office and masturbates before heading back to the house.
Marion resolves to return to Phoenix to return the money. After calculating how she can repay the money she has spent, Marion dumps her notes down the toilet and begins to shower. An unidentified female figure, presumed to be Norman's mother, enters the bathroom and stabs Marion to death. Later, finding the corpse, Norman is horrified. He cleans the bathroom and places Marion's body, wrapped in the shower curtain, and all her possessionsâincluding the moneyâin the trunk of her car and sinks it in a nearby swamp.
Sam is contacted by both Marion's sister, Lila, and private detective Milton Arbogast, who has been hired by Marion's employer to find her and recover the money. Arbogast traces Marion to the motel and questions Norman, who lies unconvincingly that Marion stayed for one night and left the following morning. He refuses to let Arbogast talk to his mother, claiming she is ill. Arbogast calls Lila to update her and tells her he will contact her again in an hour after he questions Norman's mother.
Arbogast enters Norman's house and, at the top of the stairs, is attacked and murdered by the Mother figure. When Arbogast does not call Lila, she and Sam contact the local police. Deputy Sheriff Al Chambers is perplexed to hear that Arbogast saw a woman in a window, as Mrs. Bates had been dead for ten years. Norman confronts his mother and urges her to hide in the cellar. She rejects the idea and orders him out of her room, but Norman carries her to the cellar against her will.
Posing as a married couple, Sam and Lila check into the motel and search the room Marion had occupied. They find a scrap of paper in the toilet with "$400,000" written on it. While Sam distracts Norman, Lila sneaks into the house to search for his mother. Sam suggests to Norman that he killed Marion for the money so he could buy a new motel. Realizing Lila is not around, Norman knocks Sam unconscious with a golf club and rushes to the house. Lila sees him and hides in the cellar where she discovers the mummified body of Norman's mother. Wearing his mother's clothes and a wig and carrying a knife, Norman enters and tries to attack Lila, but Sam, having regained consciousness subdues him with Lila's help.
After Norman's arrest, forensic psychiatrist Dr. Simon Richmond tells Sam and Lila that Norman's dead mother is living in Norman's psyche as an alternate personality. After the death of Norman's father, his mother found a lover. Norman went over the edge with jealousy and murdered both of them. He stole her corpse and preserved the body. When he is Mother, he acts, talks and dresses as she would. Norman imagined his mother would be as jealous of a woman to whom he might be attracted just as he was of his mother's lover, and so Mother kills any woman for whom Norman has feelings. When Norman regains consciousness, he believes that his mother has committed the crime and covers up for her. Richmond concludes that the Mother personality has now taken complete control of Norman's mind, erasing his existence.
In the final scene, Norman sits in a cell, thinking in Mother's voice. In a voice-over, Mother explains that she plans to prove to the authorities she is incapable of violence by refusing to swat a fly that has landed on her hand. Marion's car is shown being recovered from the swamp and is followed by the ending credits.

Question: Who murders Arbogast?
Answer: Mother figure