When the CA. trooper is at the front of her car, the Arizona license plate is shown. In 1989, Arizona began issuing a single license plate and that plate is always on the back of the vehicle. This is deliberate, as the remake follows the original movie shot by shot, and there the trooper checks the front license plate.
Director Gus van Sant has a cameo at the beginning of the movie when Marion enters the office after her lunch break - he is talking to someone looking just like Alfred Hitchcock.