Well, not quite geographically spot on, but the paradigm is the same.

Power over duration is power over duration, but climbing presents some unique adaptation requirements.

a) Generally a self selecting cadence that's lower than what you would use on the flats
b) A lack of "micro rests" during the effort
c) A different seat/bar angle that engages muscles differently.

The best way to become a better climber is, of course, to climb. To the point of KISS, the best way to get better/stronger/faster in non race mode is to find the longest climb you can keep in the saddle on and ride a pace which is hard enough to make conversation difficult but not so hard that you're breathing like you need a lung machine.

No hills? Head off into the wind, in the drops, at deliberately lower cadence than what's comfortable. No wind? Go into TT mode and treat every pedal stroke as if it matters...keep the load there.

And as noted, climbing at altitude is different than at sea level. Hydration requirements are higher for the same temps and if you go into the "red" the impact is much more draining and will require a longer "easy" period to recover.