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How does this one look? Or should I definitely go for the Jazz in preference to the P Bass?
I love it! I was going to ask if you could find a bass pretty much exactly like this.

It's a precision-style body with what looks like a slimmer jazz neck. It's meant to be the best of both worlds all wrapped up in one bass.
Now, you may want to take this advice with a grain o' salt. I've only been playing bass for a year, but I've been a drummer since the late 1980's. Tone and sound are not unfamiliar concepts to me. That said, here's my two cents worth
I've collected more than a few basses over the past year and the ones I keep gravitating towards are the Precision/Jazz combo configurations like this one. In fact, three out of the five that I own are P/J configs and I love them. If I want a "thumpy" sound, I turn the jazz pickup down halfway to zero. If I want a "toothy/brighter" sound, I'll bring them both up to full. From there I adjust the tone to fine-tune the sound I want at that particular moment. It's great having that kind of flexibility all wrapped up in one bass!
One other thing that I haven't seen mentioned is the subject of "which strings to use". I've tried out several brands and types (rounds, flats, tape-wound, coated, even half-wounds). I cannot stress enough the different sounds one type of string can evoke over another. I almost sold a bass of mine because I wasn't liking the sound I was getting out of it. Fortunately, I tried out another set of strings on it, and
WOW! What a difference!
I'm a big fan of the Ernie Ball Cobalts for their very "toothy" sound. Same for the DR Black Beauties. (Someone said they are just coated versions of the DR Sunbeams. Can y'all verify that? I'm just curious.) If you want a softer, more "thumpy" sound, you'll want flat-wounds. However, I don't think that'll work well for a "metal" sound. But this is probably a discussion for another thread. Back to your original question.
It's very similar to my favorite bass. I say "similar" instead of "exactly" because I haven't seen a Squier with those particular headstock decals. I'm going to guess that the "California Series" is an older model, yes? Either way, if it's a solid piece of work, it should meet your needs, both as a good beginner's bass and as something with the versatility you'll need for different sound/tone. Once you're more comfortable with how the electronics work, you should be able to fine-tune it to your heart's content.
Regardless of what you choose, I hope you find something decent that you'll love for years to come.
