In all of my cases, the dead player would simply sit and roleplay as being dead IE: not responding, being motionless, just typing '. . .' when asked a question. But I have always thought that when you die a pop up should come up saying that you've died, not a big one, in fact a small box that could be moved out of the way.
Until you hit 'ok' your body would remaint here, allowing you 'the player' to look around, and listen to further role play, and even wait for a rez spell.