It's something you can do with JavaScript but I don't think it will be looked at. The view would be email notifications (which already exist) are more appropriate, since events on this platform are not time sensitive. By contrast, if you don't notice quickly that your Showdown queue popped, then you just lost elo.
You can set up your mail client to give you desktop notifications anyway. Also there isn't going to be an app, Pokemaster has said this before.