While I am predominately a bass player, I do dabble in drums on occasion and I am married to a drummer. We had this discussion the other night of what bass players look for in good drummers and visa versa.

So as a drummer, what musical or other qualities do you like in the bass players you play with? For a drummer, what defines a good bass player member of the rhythm section?
Bass players and drummers both need to be able to keep a solid rhythm, because together they serve as the rhythmic basis of the entire band (generally). If one or the other is lacking in this department, the band will likely sound loose and unfortunate. As a drummer, I look for a bass player that can keep up with me, and a bass player that challenges me to keep up with him. That's it, really.
For me it's both the personality of the bass player and their playing as it is they have to keep solid steady rhythms and good tempo. It's all about locking in for me, if I can't lock in with a bass player I won't play as well as I should because it's hugely important to be able to. Having said that it's a two way street so some eye contact is definitely up there in my list as well as the ability to improvise and keep things interesting.
Neo Evil11
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I like to listen to the bass player and expect vice versa... that's pretty much it for me, at least. Elaborate note choice and all that's cool but it's secondary in importance to groove and/or pocket for me. I like to improvise a lot so I like it when the bassist won't get lost during the times that I do and can maybe even improvise some stuff in response.
One of the most important things about the relationship between bass and drums is that the bass drum and bass players parts meet - usually to emphasise parts.
Drummer of 7 years, Bassist of 5 years, Guitarist of 4 years.
I like to jam with a bass player who can really groove in the pocket and then mess with that and take it out into uncharted territory. Typically I find bassists with funk experience are best at that. I also like it when they can bust into a solo, where I can just play a simple beat and listen to them, almost as if I was an audience member for a moment (musician's shouldn't just be playing for the audience, they should be playing for and with each other). I find jazz guys are the best at that.

But all I really need in the context of a serious band is a guy who can play the written material, nothing more special than that.

edit: well, they gotta play it in good time and with good tone, but that should be a given for any musician...