Does Monk Stay a Detective?

Possibly due to this, as well as the many cases Monk has solved over the years, he is reinstated as detective first class by Stottlemeyer in the season 8 episode "Mr. Monk and the Badge".

Does Monk quit being a detective?

A week later, Monk is further disappointed when he is removed from desk duty, and assigned to a homicide detail in another part of the city. His new boss, Sergeant Danny Weaver, is a good cop in his own way, but he's no Stottlemeyer; neither is his new partner, Detective Lewis Doyle.

Does Monk become a private investigator?

Monk crawls around the edge of the boat, then quickly consults a "swimming fundamentals" card he keeps in his wallet, before jumping overboard. ... For all his brilliance as a detective, Monk just isn't cut out to be a private investigator. A few days later, Stottlemeyer moves his things into his new apartment.