I mean one could argue that started with the normal run before the Timeless child. Maybe not immortality but a crazy resistance to injury.
The 10th Doctor fell from a ship high up and landed in front of the Time Lords reborn and while injured was not crippled.
The 11th Doctor fell from orbit and survived. They said something about the suit helping and fixing him, but still he had to survive it for that to matter.
Resistance to hard radiation and other things too. The Doctor has gotten more durable in the end.
I mean for his enemies he is practically immortal anyway.
Since the regen rules are now all over the map, just easier to say that he can now heal from nearly anything!