Bethesda didn't buy id, although Bethesda will be publishing id's upcoming games.
Bethesda hasn't said, and we don't know what their next game will be so it's hard to speculate. However I would say it's extremely unlikely that Bethesda's next game will use id tech 5:
By the time id joined Zenimax Bethesda had almost certainly started work on their next game. If Bethesda had already started working on a new engine for their next game developement would have been well under way by the time id joined - it is possible that Bethesda had already decided they wanted to use id tech 5 but if so this decision was made without the benefit of their future relationship. I'd also say that given Bethesda's experience working with gamebryo it seems more likely that they would have stuck with them.
Alternatively Bethesda may not have started working on a new engine and may have decided to do one more game with the Oblivion/Fallout 3 engine. In this case their next game still won't use id tech 5 obviously but after that they will be in the market for a new engine and id tech 5 will probably look like a good option. It is possible that Bethesda will stick with gamebryo given their familiarity with the engine and its focus on mutability . . . . but the financial benefits of 'purchasing' an engine from your sister company (and the extra exposure they'll get from this) sounds like a strong arguement for id tech 5. The real hurdle, I imagine, will be whether id tech 5 can be modified to fit Bethesda's requirements.
Will the next game on the gamebryo engine have perfect player animations? That is, the laws of physics apply to the walking player? That is, if I move in a circular motion while looking in one direction, my feet won't slide all over the ground, thus making me look like the player from Quake 1 or Hexen II?
No offense, but the animations in oblivion svcked, and I thought that in fallout 3, it would have been fixed, but it wasn't, so I would have put blame on the engine itself.
I also suggested id tech engines, because the player animations, well at least the strafing left and right has decent animations in id tech 2, but in oblivion or fallout 3, when you are strafing, the animations state that the player is trying to move forward, not sideways, and that is important if there is to be a third person camera angle.
I believe unreal and source has decent strafing animations too. If it isn't the engine, then it is probably the animators.