I'm lucky to own a 1968 Fender Precision bass. It's one of the holy grails of bass tone and has been heard on countless albums over the decades. James Jamerson perhaps, did more than anyone to popularise the P bass tone: thick, punchy, and prominent in any mix.
One thing I'd always wanted to do was put together my own bass from parts and that's what I did with my Warmoth P bass. Watch the video below for the lowdown on parts, electrics, and hardware. If you ever get the chance to try something similar yourself, I highly recommend it.
The Warmoth is my backup to the real deal but to be totally honest - for me - nothing beats that vintage Fender. There's something about the way it feels as well as that sound that is difficult to replicate in new basses.
However, see what you think about the tonal differences in the video below. When you mod your bass, you can be in charge of what you want your bass to sound like. If you have a decent body and neck, there are so many upgrades you can make to your bass.
Here's a video with some pick, fingerstyle, palm muting, and slap comparing the tones of the Fender Precision against the Warmoth bass guitar.
You really can get very cheap great sounding basses these days. If you're into recording, or want to play in a band, I don't think you can go far wrong with a Fender Precision. Leo Fender got it so right so early and we're all the beneficiaries of his genius.