I'll move this over to the Game Opinions & Ideas board, as you are suggesting a change to the game.
I'm not sure if we'd be able to add it without the source code, but I love this idea. As someone whom is constantly tabbing in and out, I can see the assistance of an audio cue to nudge me to my senses when multitasking. It may be tricky or prove a bit counter-productive with a "ping!" each time a message is posted in a public chat, but one that notifies you when whispered would be helpful; perhaps, even, one that allows you to be notified when a specific person is speaking. (This one could help if you're trying to keep an ear out for a friend's message, as you referenced.) Of course, you would probably have the ability to toggle it on and off, too.
Cute idea. c: