I think you will find that there have been`rescues` carried out in the Himalayas at or about that altitude,probably by AS Llamas or Mils,and an AS350B3,landed atop Everest,briefly,a couple of years ago.However ,the problem is mainly the approach/departure paths to the helipad,and all the topographical wind effects.It would also depend whether you were intending to drop-off `stuff` or pick-up,whether it was an isolated pad,ie on a pinnacle,or just an aiming point in a cleared area,as this would determine hovering `outside/inside` ground effect power requirements.I`ve been to 20kft in a Gazelle a few times for certification,and you are in a `coffin corner`,in that if you slow down below(guessing) 20 kts,you will run into loss of tail rotor control,and lose height; if you go for more speed(guess again) say 50 kts ,the rotor can be suffering compressibility effects;if you still can apply power to climb ,that margin can become very narrow;then of course if the windscreen is icing up inside,and you turn the heater up,you rob the engine of power so you`ll descend;and of course you need oxygen as well...
All these effects are similar to fixed wing a/c at 60k+,but magnified in a helo
,and then you have the descent to consider,as you can`t just dump the collective,as the RRPM will be off the clock in a second,into compressibility,and instability......I haven`t `googled` the OGE perf. of the K-MAX,Chinook,or some of the Kamovs,but they may have an edge at altitude over conventional helos....Hope that raises your spirits ....Syc
Anyway,a Harrier couldn`t do it,eh,John...