I don't overly love the P bass. There is a pattern, though... I have a Fender American Standard, a cheap copy, an Orange O-Bass which is sort of a P bass and a Guild Starfire I, with the split pickup (so kind of a semiacoustic Mustang?). Right now, I dunno why, maybe it's the strings (Rotosound rounds), maybe it's the pickup (EMG GZR), but my favourite of these is the cheap copy. Followed by the Guild, especially with flats. The Orange doesn't really sound like a P (dunno why, maybe it's the okoumè body or the slightly different pickup position). My American P isn't quite there, although it is certainly the best of the four construction-wise. Right now, with DR Fat Beams and the stock pickup, it sounds too bright, and I actually have to cut the high mids quite a lot to get it in the whereabouts of where I want it.
The truth is that the split pickup in the middle happens to be one of my favourite pickups. I have tried Jazz basses, countless times. I like the way they sound when other people are playing it. I hate it when it's me playing it. It sounds thin, wimpy, and I hate the bridge pickup soloed. The neck pickup is OK but still not a split pickup. On the other hand, I quite like PJs, provided that the two pickups sound balanced together. I feel that the bass that best pulled off a PJ was the older Yamaha BB line (not the current models).
And anyway the bass that I most like playing is not over of the aforementioned. It's my cheap Chinese Höfner Club with a minihumbucker blade pickup at the neck, soloed, and with none of that tone filtering bollocks (I made it into a regular Volume - Tone affair).