Bitcoin is never truly anonymous. If you buy BTC from someone/somewhere that needs ID, they know the address you buy the bitcoins to are owned by you. Blockchain.info makes no difference on that.
Thank you @DarkStar_
I will not be buying bitcoins. Somebody owes me money and will be paying via BTC and I will be using those bitcoins to pay for web hosting etc. I will be using one or two layers of anonymous VPN to login. VPN + Tor would be better though and there was a discussion on reddit that said blockchain.info banned Tor usage -
https://www.reddit.com/r/Bitcoin/comments/3lzfp9/blockchaininfo_doesnt_work_via_tor_at_all_not/