Updated review (24March2015)

After battling it out and not having push notifications for almost 10days, Tapbots have finally been able to solve the problem. Everything is working as it should. Im so glad I didnt have to resort to using an alternative Twitter client, Tweetbot is like the caviar of Twitter clients, if you like caviar.
I dont know if its an API limitation, but I wish if Tweetbot included a feature for new follower requests for private account users. Other than that, great app!

PROGRESSIA about Tweetbot 3 for Twitter. An elegant client for iPhone and iPod touch, v3.5.3