Is there a kind of application like Twitter based on the blockchain?
I think the blockchain would be suitable for such an app. Steemit and Qora have shown that a decentralized blog system works well with the blockchain. So why not a decentralized messaging app, too.
Yes, there is:
twister
Peer-to-peer microblogging
twister is the fully decentralized P2P microblogging platform leveraging from the free software implementations of Bitcoin and BitTorrent protocols.
http://twister.net.co/