How "custom" do you want to go? Some people want something that looks one-of-a-kind (or made with rare exotic materials), while others simply want exceptional sound and playability with more-or-less traditional styling. And since bass tone varies over a very wide range, you need to consider which custom builders are renowned for delivering the sort of tone you are looking for.
I'm a Fender aficionado, so I'd want a custom bass to look much like the familiar P and J styles but play and sound just the way I want it -- not necessarily "better" (that's a very subjective determinant), but personalized for my particular tastes. If I could get that in something that is easily reproduceable/modifiable (in case it breaks or is stolen), so much the better. And if I could get it at a somewhat affordable price, better still. Fender Custom Shop might make what I want; or Sadowski; or any number of other builders. For you, something entirely different might be absolutely necessary.