There is a fairly popular chrome extension called StackNotifier that let's you get notifications in chrome about answers to your questions, incoming questions etc. Chrome supports web push these days so perhaps it could be implemented natively on the site? That way it would work in Android as well.
This feature is partly supported already.
The mobile apps show notifications like the site does. Whenever there is a notification available, it will pop up immediately.
Chat already supports desktop notifications though the HTML5 Notifications API. And it works well.
I don't think it is much work to incorporate this into the main site and I would love to see this option available. When I'm active I go to my browser to see the notification after my phone (which is on my desk) told me there is one!
For mobiles, I think the app is good enough: you get push notifications for anything important.
However, for the web version, you can use my Stack Exchange Desktop Notifications userscript.
It displays HTML5 notifications whenever you have:
- new answers
- score changes
- new comments
- new inbox notifications
Most of those relate to a question you have open in a separate tab, but the 'new inbox notifications' is mainly what you're asking for!
Visit the Stack Apps Page for more information on usage and installation instructions.
Note: This script requires you to have a tab open with an SE site/question!