We use pressure altitude for a reason in that it correlates to aerodynamic performance, which is critical at each end of the performance envelope. So I won't go so far as to advocate using absolute GPS altitudes, no matter how accurate, as filed/directed altitudes for flight.
Now, that said, the idea of using such GPS data, assuming it is EXTREMELY accurate (i.e. +- about 6 inches), presents some intriguing possibilities as another source of data for pilots to determine attitude. I'm not an expert on this, but when flying pitch/thrust in an emergency, I doubt GPS has the ability yet to provide information with the precision required. However, it does have the potential to present an indication of extreme vertical speed, which would be valuable data to have in an upset. If we can get GPS data accurate enough to provide attitude data accurate enough upon which we can rely, it would be a good tool to have in the toolbox.
IOW, a thoughtful idea, but one which I question whether the tech is yet up to the task.
I'd be pleasantly surprised if someone can chime in and prove to me that my assumptions regarding accuracy are incorrect.