The main purpose of alternative tunings on guitar is to allow for unusual chord voicings, especially to have a particular “mood sound” from the open strings alone. This applies also somewhat to other instruments, in particular scordatura on cello or violin, but doesn't really make much sense for bass because open strings are so low-pitched that using them for any chords gives little more but musically useless sub-rumble. Bass is usually played mostly monophonically, and for that switching between tunings accomplishes little but messes with muscle/scale memory. The only alternative tuning that's really common is drop-D, and basically its only purpose is to extend the bottom range on a 4-string bass. Same with drop-E♭: you'd use it if a piece benefits from having a low E♭ but you want to change as little as possible about the instrument or effects.
That said, for virtuoso bassists alternative tunings do become interesting: some use chords on unfretted strings a lot, but not so much on open strings as flageolett harmonics. There's a lot of interesting options here, in fact far too many to fit them in an SO post. There's a lot you could just try and experiment, which is probably how most guitarists go about this as well. For example, the one piece I wrote for scordatura guitar is heavily based around the mysterious-misty sound of the sus4♭69 chord, in this case D-G-♭B-E. It's used mostly in harmonics, which is what you'd probably also do on bass.