Twitter has announced that soon you will be able to receive pop-up notifications when someone interacts with your tweets or send you a direct message. Not only that, you can also do a quick action in regards to that notification, such as replying a Direct Message on the pop-up window.
The pop-up will of course be quite subtle, so it does not open up a new browser window on top of your face like the old days. The update should be rolling out to you soon and you have to use twitter.com on your browser.
To opt-out (opt-in), you should check your Twitter’s profile settings (that Gear icon on the top). When the update has come to you, you should see the Web Notification settings like this one:
For the past months, Twitter has been working hard with new interface changes, features, and improvements (including frequent mobile updates). It’s great to see them supporting and taking the platform seriously.