The standard expedition stove for extreme conditions would be an MSR XGK. You will likely want to bring a pair of them, along with a repair kit, on the assumption that due to the cold or poor quality fuel you'll break a pump or need to make other repairs. Now, you may be thinking "what are all those people doing with canister stoves on Everest/Aconagua/Olympus Mons?" Well, it's true; many people use canister stoves like an MSR Reactor or WindPro with an isobutane/propane mix at altitude.
There are a few not-so-obvious reasons behind why someone might turn to a canister stove, most having to do with the altitude. First, being above 6000 meters means having less oxygen to work with. This means your stove will likely burn rich (yellow flames, less complete combustion). The lower air pressure will help offset the colder temperatures which normally present issues for canister stoves. It also means your brain will have more difficulty with complex tasks, so the simpler the stove, the better. Finally, it means there are a limited number of places you might be: the Himalaya, Andes, or on Denali, and that may affect what fuels are available as well as the likelihood of being forced to cook inside your shelter.
Fuel Availability:
If, for example, you are in the Himalaya you may only have kerosene available, and not white gas (naptha). Kerosene will have more issues with the lower oxygen levels at high altitudes (particularly around carbon monoxide production), and will also make a bit of a sooty mess (of course, so does white gas). It is also a fairly dirty fuel and may clog your stove, necessitating more frequent repair. Regardless of fuel type, if you bring a liquid fuel stove you should also being a repair kit.
Cooking inside and carbon monoxide risk:
While every stove will come with clear warnings about how unsafe it is to use it inside an enclosed space, this sometimes becomes a necessity in mountaineering. This warning is mainly out of concern for carbon monoxide (CO) poisoning, but there is also the risk of catching your tent on fire. (Silnylon is particularly vulnerable to fire, and flareups do happen.) Thus, lighting a liquid fuel stove with its usual priming "fireball" becomes problematic. (You really don't want to use a white gas stove inside a tent!) If in-tent cooking is a likely possibility, you should have a good grasp on how to mitigate the risks of CO poisoning through proper ventilation and fuel management. In theory canister fuel should produce less CO than liquid fuel, but this is also affected by stove design and the pot used. Often CO production can be also reduced by raising the pot about an inch above its normal position, to allow for more complete combustion.
Know that this is an easy way to end up dead, so you should know what you're doing before trying it, and only try it if conditions are such that cooking outside your tent would also make you dead! I suggest reading this thoroughly as well: http://zenstoves.net/COHazard.htm
Cold temperatures:
Regarding the temperatures, you still want to use a winter fuel mix for canisters. Usually this is a 30% propane, which vaporizes around -40 degrees at sea level, and includes isobutane (boils around -12C) as well as the traditional butane which boils around 0C. However, higher altitudes mean lower air pressure and thus the fuel in a canister will vaporize (boil) at lower temperatures (just like water does). For example, the -12C vaporization point of isobutane drops to -27C when at 6000m.
It's often described that in cold temperatures only the propane will burn off, leaving just the butane. Gas mixtures do not actually behave that way; they instead form a blend: a 70/30 mix of butane and propane will have a boiling point atound -18C. The propane will vaporize at a greater rate, however, and the "boiling point" of the blend slowly rises, eventually becoming that of pure butane (because that's all that's left). Using a stove that allows inverted operation will help avoid this, as you are burning the canister fuel in liquid form. More reading at http://adventuresinstoving.blogspot.com/2012/04/gas-in-cold-weather-myth-of-fractioning.html
If you do use a stove that allows inverted operation, don't just flip the canister over. Stoves that allow this route the fuel through a pre-heat tube that passes through the flame in order to vaporize the fuel for burning. This means you need to let the stove run with the canister upright for 30 seconds or so until the tube can properly pre-heat. Be careful when turning over the canister; sloshing fuel can easily cause a flareup!
Still, you will often need to take steps to warm the fuel. Always remember that too much heat will make the canister explode. You should start with always keeping at least one fuel canister warm against you or in a sleeping bag. An insulated base for the stove also helps with both temperature and stability. To maintain output while cooking, you should have a stove with regulated output (i.e. as the fuel cools it opens the throttle to keep the pressure up). Setting the canister in about an inch of water should also provide a temperature buffer to help keep the canister from going below freezing. Warning: Extreme Danger Here! Some people use more interesting methods to heat the canister, including hammering flat some thin copper tubing and using it to bring heat from the flame back to the canister or using a handwarmer.
If you are applying heat to the canister, remember that the canister should never be too hot to touch, and you should always have a (nonfrozen) hand on the canister to make sure of this. ("Too hot to handle" is usually around 42C; expect the canister to have a "core breach" around 50C, and realize that dents and low air pressure at altitude may cause it to rupture before it gets that hot.)
High winds: Stoves like the MSR Reactor are particularly good for handling wind, but with other stoves cases you'll likely want a windshield. With canisters you are often cautioned to make sure the shield does not cause the canister to overheat, but in cold weather this can be helpful. Just remember to monitor how warm it gets!
Lighting your Stove:
You may find that piezo ignitors and other fancy lighters stop working due to the cold or altitude. I suggest having multiple starting methods; a simple mechanical Bic lighter kept warm close to your body is good to start with, and a ferrocerium rod + striker makes a good backup. Waterproof or stormproof matches make a good third choice.