We have written about it before, but the first Twitter client with push notifications have now become available.
Beezz, which is free, provides push notifications for your time line, mentions and Direct Messages, and use both Live Tile and Toast Notifications.
So far in my testing the client worked quite well, although some users have reported the notifications do not work extremely reliably. However, as the client is free, I will take that over nothing at all, which is what the official twitter client offers.
The software can be found in Marketplace here.