The possibility of implanting memories could distruct the idea of the persistence of the person in whose mind the memory has been implanted.
In fact, if in the mind of person A all the memories of person B were implanted, A would no longer be just A. She would be affected by B's experiences, tastes, etc. Hence, memory is not enough to establish a lasting person –at least if one wants it to face such extreme cases.
But, I wonder, why should one not accept that A-after implantation is a different person from A-before it? In fact, they share the same body –but this is no argument, if one does not share a reductionist viewpoint. They also share A's former memories. Is this enough not to admit their being different? An interesting literary (and medical) case would be that of Robert M. Pirsig, the author of Zen and the Art of Motorcycle Maintenance, where the case of a person undergoing an Electroshock is depicted. The author claims to be not the same person his wife and children would recognise as him –notwithstanding his sharing the same body and many memories with him.