Although it's not quite the same as a native app, we already have built in support for desktop notifications on all browsers that support them, including some older WebKits.
There are still a great many organizations that are using XP. It's a reliable and battle tested OS. Ignoring these users wouldn't be a good idea, I don't think.
This works in Chrome: http://jsfiddle.net/dandv/wT26x/1/
I think Safari has a system for this too: https://developer.apple.com/notifications/safari-push-notifi...
Firefox: https://developer.mozilla.org/en/docs/Web/API/notification
CanIUse stats: http://caniuse.com/notifications