No one really uses steemit itself for anything other than to post I think, personally I use busy.org as it has notification built in , and always have steemworld.org open which is also going to release mentions etc soon.

As for desktop notification I think you can use this https://steemit.com/steemit/@revo/steemnotify-me-steem-notifications-in-your-browser-update

But I am happy with my busy.org , steemit and steemworld.org combo.