Factual error: Near the end of the episode, Jackie Cooper fires a bullet into his hotel room from the balcony using a revolver fitted with a silencer. Hayward also uses a revolver with a silencer to kill Stone. This is a classic Hollywood error, as silencers do not work on revolvers (unless noticeably modified), due to the gap between the barrel and the cylinder. They also don't make gunshots nearly as quiet as shown.
Factual error: Columbo parks across the street from a house where a murder has occurred. The address is given as 1278 Fairfax Drive. Columbo investigates and returns to his car. A house number, 400, is painted on the curb. The house across the street from 1278 would have an odd number, probably 1277 or 1279. It would not be 400.
Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73