Did Roger Chillingworth change his name to Roger Chillingworhth?
I wrote in an essay that he did because it says "old Roger Chillingworth, as he was hereafter to be named" (or something like that). That could mean that "from now on in the book we will know this guy as Roger Chillingworth, as it was the introduction of his name. Also, wouldn't Hester's name be Hester Chillingworth and not Hester Prynne if Chillingworth never changed his name?
Yes, Roger Chillingworth did assume a false identity. Prior to doing so, his last name had been Prynne. Had Chillingworth been his real name, Hester would not have been Hester Prynne, but Hester Chillingworth.
There were a few reasons for his choosing to go by a false name. For one thing, Hester's affair and the resulting birth of Pearl, an illegitimate child, would have been cause for a great deal of humiliation directed at Chillingworth had anyone known his true identity. In addition, Chillingworth's alias allowed him to infiltrate society and pursue his enemy without being detected by the man. Because of his fake identity, Chillingworth was able to play the role of an innocent citizen and make others see him as he chose.