I fear this altimeter idea may be scuppered by local variations in the Earth’s density (it’s not an exactly uniform sphere of rock). Or maybe that just means the clocks could be great density-mappers!

It is easier to measure density with gravimeter than measure gravitational force directly.

Satellite, ACES, was launched recently that uses atomic clocks to accurately measure Earth's gravity field.

Are you saying they would be relatively inaccurate?

I had the same thought, but I still want one!

On second thought, you need a base station on the ground to tell you its time for comparison anyway, so if that base station is nearby the density thing should mostly work itself out