Erikstormtrooper said:

Here's a thought:

If Sidious gives Anakin his new Sith saber, maybe we can assume Anakin just keeps both. But the old blue one stays out of sight until the very end of the Mustafar battle.

Right before Anakin says "you underestimate my power", there's a wide shot of him standing on the floating platform. If you reverse the sequence of these 2 shots, in the wide shot you can then add an ignition of his blue saber in his left hand - so he'll have a saber in each hand. He'll make the jump with both sabers and will drop both when Obi Wan cuts him down. When Obi Wan picks up the blue saber, you can even rotoscope in the red saber sitting nearby. (Maybe that can be a symbol of Obi Wan clinging to the good side of Anakin while rejecting the bad).

This is an interesting idea.

However, I don't really understand what this whole fuss is about. Sidious gives Anakin a new lightsaber and then, after ROTS, Obi somehow gets his hands on Anakin's old one.