No, not all communists are Marxists and the predominant tradition, anarcho-communism, is not post-Marxist but derives from a different set of authors who opposed Marx during his lifetime. [1]
Thank you. Some of the comments in this thread are so bizarre I have to wonder if it's down to US educational systems placing everything that's not Reaganomics into a basket marked "Communism BAD".
1. https://en.wikipedia.org/wiki/Communism#Other_types_of_commu...