There are very few reasons for needing that level of anonymity -- and none of them are good or honourable. Two purposes come to mind: sock-puppetry & trolling.

There are no legitimate reasons in having two or more comments in the same thread, by the same anonymous source, disassociated from each other.

For all other purposes, the ability to hold a conversation, anonymously, but with the continuity of the exchange unendangered by crosstalk with other AMs -- accidental or malicious -- can only be a good thing for all parties to the conversation.