I play a 4001 and a Precision. 4001 is the earlier version of the 4003, but in general terms the 2 Ric types are very similar.
I actually find 4001 and P fairly interchangeable. The Precision is far more tolerant of enthusiastic picking. If you have a reasonably low action, with the Ric it's dead easy to catch a string against a pickup or knock it off the bridge saddle. A P bass, you can pack the neck joint to alleviate that trouble. Can't do anything about that with the thru' neck Rickenbacker. Some say you can't slap a Ric - yes you can!
Position wise the Ric "Bridge" or "Treble" pickup is more like the Precisions sole "mid" pickup. Especially without the bass cut capacitor fitted in earlier Ricks (switchable nowadays I think). Having Ceramic magnets, my Ric pickups are a tad bright which exaggerates string rattle with new stainless rounds. I've found Nickel wounds better from new on my Ric.
The Ric neck pickup, together with the pickup selector I find useful for quick switching between background/foreground work. Not impossible on a P, you just adjust playing technique and/or twiddle a control. Don't dismiss the P-J type of Bass if you want that kind of versatilely, but they seldom have a selector switch.
My Ric has a 2 channel "stereo" output option not possible with a single pickup instrument (actually, it is, but that's a different thing altogether).
It's worth remembering that one famous Rick player of yore, Yes's Chris Squire, used a '51 type Precision before he got the 4001 and kept it as a standby for when the Rick needed work. He sounded like himself whichever.
The modern P bass is possibly more comfortable than any Ric. Mine is, but my Rick is actually a non-reversed R/H one with an extra strap button on the short horn to hang upside down, so I'm missing some of it's comfort sculpting. The P is left handed with reversed strings and is so comfortable, I hardly feel it against me.
A big plus for a L/H Fender if you're reverse stringed - reverse the nut and adjust the saddles and it's ready. Can be put back anytime. A Ric would also need a new Nut. If you're normal stringed, there are plenty of l/h Ric's out there.
Don't dismiss the P-J bass variant. That's where a Jazz bass pickup is added in the bridge position. My Precision is being modified for that right now, with a selector switch and stereo jack.
_________________ You can call me Jim
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