Today, Twitter announced real-time notifications on Twitter.com for direct messages, replies, favorites, retweets and follows. When someone engages with you, you will get instant interactive notifications. The features is rolling our and will come to you in coming weeks.
See the example notification in the snapshot below:
If you are using Twitter on Twitter.com, you will receive these interactive notifications. And you can also reply, retweet and follow the new follower direct from the notification.
To change the notification settings, go to your Twitter settings page. Here, you can choose what type of notifications you want on Twitter.com.
“Web notifications” settings section will only be there if you have received this feature. If it is not there, wait for few more days.
Company tested these notifications with some users back in January. In past, Twitter has rolled out few cool features like photo tagging and new design.