I personally chose Fender, because for what I play, I find my USA Jazz to be more fitting and versatile. However, I sure would love a Gibson Thunderbird to replace my Epiphone copy, which has been a real work horse for about 5 years now.
To be honest you need to give us more information for us to be able to give you any decent reccommendation. FENDER VS. GIBSON isn't really enough, it'll just devolve in flame wars and personal bias as it already seems to be.
Really, you need to get down a music shop and get your hands on as many quality basses as possible. I mean just because your choice of Bass may not be conventional, doesn't mean that it would sound brilliant in the projects you play. I chose a Jazz because I love the sound, it's versatile and because the one I played in the shop felt amazing in my hands; love at first play. But if it had been a cheaper Squier or a Warwick or a Rickenbacker or any other bass that felt and sounded that good to me I would have gotten it; I didn't just limit myself to one particular brand. Don't make that mistake, or you may pass up your perfect bass just because it has the wrong name on the headstock...