Unfortunately, the transfer of characters between accounts is not possible. Using the same account across generations within a console family (360 <> One, PS3<>PS4) is possible, but there is no means of moving characters across console families.
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is it possible to transfer characters between ps4 accounts that you both own?
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No, data transfer between accounts is not possible, even internal to a console family.
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How was it possible for my friend, who was originally an Xbox 360 user, was able to transfer over to a PS4?
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He wasn't. He had the start over when he started playing on his ps4. He's lying if he told you otherwise.