I really, REALLY wish Bungie would allow you to transfer characters across different platforms.
I have the game on both PS4 and Xbox One because I have groups of friends that play on both, but my Xbox One character has way better gear because that group plays more frequently, and I can actually stand to play Crucible on that controller*.
I'd love to just be able to use the same character on both platforms, and considering all my character data is stored on Bungie.net anyway, I'm not sure why it's not possible.
*Nothing against the controller, it's a great controller. Purely personal preference, I'm more comfortable with the 'feel' of the sticks, triggers and ergonomics of the Xbox One controller for fast-paced multiplayer games.