Would it be possible to receive a notification when one's opponent has types in a new message in the Chat box of an Online Chess game which is still in progress?

The notification could take the same form as for a normal message.

I believe chess.com already does this? Or maybe it's just for finished games?

I meant messages in the Chat box from the opponet while the game is in progress, not the comments underneath the board.

Steinar is correct: for a little while after the game is finished, if someone types something in the chat box, the opponent get an alert for the game - even though the game is actually finished.

But it would be neat to have it during the game - as you propose.