I don't have the brain power right now to think of a convo with myself so I'll leave it up to someone who does.
If you met a replica of yourself how would your conversation sound?
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Seeing as this person is me, we would most likely acknowledge the other's existence and briefly attempt to explain the other's presence, all the while speaking in unison until a silence falls as we both realize that nothing we say will not have already been thought by the other.