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Re:Twitter’s Chief Technology Officer to Leave Company by scholia on Tuesday December 20, 2016 @09:29PM (#13226356)
You can already reply to yourself and Your handle doesn't appear, does it? At least, it doesn't if I use the web interface.

Otherwise, this pic will give you a good idea of the changes Twitter is making to the tweet structure...

http://www.insider.gr/sites/default/files/57e01f4edd08956b5e...

Re:Twitter’s Chief Technology Officer to Leave Company by mwpmaybe on Tuesday December 20, 2016 @11:19PM (#13226757)
Even if it does appear, you can simply remove it and Twitter will preserve the reply/threading regardless.