Your "five minutes ago" consciousness would continue along in the original body, and would have the sense of being the same person. The copy would have that sense as well. If both copies are aware of the copying process and which one is the copy, then they could agree that the original is the original "you". But if the copying process destroys one and makes two, or in some other way makes it impossible to distinguish original from copy, then both people could legitimately claim to be the original, and to have a continuous conscious experience of being the original.

That'd be kind of messed up.