I just got a 96' 328i e36, and I hear a slight buzzing sound from the center console when I shut the car off. It stays for a bit and then goes away....

Any help on what it could be?

There is a small vent in the front of you A/C control panel. Inside of that little vent is a fan, to draw in air for the interior air temp sensor. (for automatic operation) Usually that fan with get a bunch of dust and crap stuck in it. You can see this without removing the control unit. If this is whats making your noise, you can just blow it out with some compressed air.
If you do end up blowing out the fan/vent. remove the control unit, so all the crap doesn't just go into the electronics.

okay thanks, ill do that, so i have to remove the whole A/C unit?