DING.

[Dreadpiraterobertzzz]

Okay, this should be an easy one.  When you get a new private message, it should play an audio alert.   Almost nobody responds to private messages within 15 minutes, because they just go at the far edge of the chat window with  no notification and we're already used to ignoring several tabs there that aren't immediately useful.   Private messages are often of immediate interest.   

AOL solved this problem in 1997.  We can catch up!   There can be a checkbox to disable it in settings if there are people who just can't handle a notification sound being added to the game.   It could sound like the 'captain on deck' bosun's whistle signal!   Or the watch bell sounding.  Or a signal gun going off.  Or a sailor shouting.  Or whatever!   But some notification when a new message arrives would be awesome.

[Papillon]

My phone annoys the heck out of me because for some reason, it continues to turn back on all those notifications that I keep turning off over and over again.  So annoying.

However, I completely agree with OP.  I can't imagine why something so fundamental as chat notifications are not in-game.  Why not devs?  Basic functionality and quality of life.

[Macjimm]

Great idea.  Absolutely no downside (its optional).  I am skeptical it will be implemented.

[Karvala]

Yes, another vote here.

[Teutonic]

Yes