As an amateur keyboard player, I am used to comping with (or against) a guitar solo, sometimes staying out of the way, other times harmonizing against it, or even just "responding" to its calls. On the other hand, I have absolutely no such experience with a bass solo. What are best practices in such a scenario? When should I stay out of the way? When should I take over the original bass line with my left hand to make sure that the groove doesn't just stop? When should I respond back to the solo? Frequencies can get pretty messy very quickly down there, so I'm hesitant as to whether I should even keep playing in the lower range at all.
Since the solid bass playing will stop during a bass solo, only coming back in during the last couple of bars, maybe, the drums often continue. That is to keep a rhythm going, sometimes even against what the bass is doing to solo.
On occasions such as that, I may put stabs in, usually on beat 1 of every four or eight bars, or where there's a fundamental harmony change. If he's soloing alone, he may or may not need extra stabs to keep his timing accurate. Sometimes stabs to keep the thing in shape for everybody, sometimes when I feel the bassist is also going to put 1 on beat 1, sort of to underline it, and present a togetherness.
Don't bother trying to play another bassline underneath, he won't appreciate it, and it'll all get mixed up and muddy. keep it all simple, and providing the bassist stays in tune and time, get ready to greet him with a triumphant chord as he comes out of his solo finally. then wait for the applause (!) before wowing the audience with anything too flashy.
When I'm on bass, I really prefer nothing at all from anyone except drums, but that's improv. If it was to be a proper worked out solo, then anything could be rehearsed into it, that's rather different.
Note: this answer does not presume there are no female or other bassists...