The bi-regeneration was caused by the half-human 10th Doctor trying to save his own life. He figured he could use the ‘space-time’ event of a regeneration to ‘lock onto’ the Doctor and cross over from the alternate reality and to ‘take over’ the Doctors regeneration and transfer into his body and get the regenerations. BUT… it was the Whitaker Doctor’s regeneration he merged with. That’s why 10th’s face returned. It was just an imprint, but no substance.
Then he tried again, but locked on to the 14th Doctor. This didn’t work so well either and it caused the bi-regeneration with 2 of the “same” Doctor trying to occupy the same place at the same time.
But the half-human doctor ‘bounced’ and landed somewhere else in the correct reality. Rose Tyler knew what had happened and that the half-human Doctor was gone from her reality. So she tried the same thing to come to this reality and save the Real Doctor. But when she arrived in the correct reality she caused the REAL Doctor to ‘bounce’ and land elsewhere.
So it’s now a rush for her to find the Real Doctor before the half-human Doctor can find him. Of Course the half-human Doctor also knows of Donna and in the end he extracts the remains of the meta-crisis Doctor out of Donna and her daughter, making them normal humans again.
This makes him a full Timelord again, but no regenerations. And in the end the half-human Doctor becomes the Valyard. He steals the 15th Doctors clone TARDIS and the 14th Doctor Regenerates into a different 15 Doctor and Heads off after him.
Rose either goes back to her reality, or dies… Or she goes off to screw-up Micky and Martha’s relationship because she’s a horrible person. haha
Personally I think this is 1000x’s better than what RTD will create.

