Continuity mistake: Viveca's beauty mark switches cheeks several times between shots while she talks to her assistant. (00:07:45)
Continuity mistake: Dr. Murcheson is not wearing gloves when he scrapes the women's faces. Then he goes into another room and removes a gray glove from his left hand.
Continuity mistake: Just before Viveca murders Martin Sheen, she scribbles an amount on magazine-her nailpolish is red. Seconds later after the murder with the microscope, Vera's nailpolish is a completely different color, mauve.
Continuity mistake: During his poolside talk with Nelson Hayward, Columbo carries a large cardboard box under one arm. His coat tail is hanging straight at his side in the master shot, but when it cuts to another angle, the coat is draped over more than half of the box. (01:15:00)
Continuity mistake: Just before the end we see a slide of Columbo looking at the lamp, but in the next shot of Columbo the slide is reversed.
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