I do not like Fender Jazz basses. I like that model of some of the other builders, but.... yawn. My main bass now happens to be a Roscoe Century 6j. The only thing remotely jazz about it is the aperture and placement of the pickups are similar. Style, construction, etc., are vastly different. The kicker: it has the capability to get an awesome P tone.
If it wasn't for that bass, I would choose P-bass. Even Fender P-basses, I love. Even better, ideal even, would be a PJ, non-Fender. That's my #2 currently. I could easily do everything I need to with a nice PJ 5 and not miss anything.