"They can now allow processors to work at full power"
Oh, because right now I'm only using what, 75% power ?
This may be an interesting idea, I don't know, but that sentence has some powerful cognitive dissonance. An ultracapacitor will allow burst mode functionality for sure, but only for as long as it holds energy to do so. And I don't suppose we're talking megawatts, here.
Once said capacitor is empty, the processor (and the rest of the power-hungry elements of which there are a few more than just that one) will have to rely on plain old socket power, or power from the rest of the battery.
And the ultracapacitor will not recharge itself faster than current can get to it, no matter what magical graphene/unicorn horn you put in it.