I know that cold weather can be dangerous, especially if one is not in proper winter wear. But I came across a video whereby a person is seen digging ice for water and bathing in an icy, mountainous environment. What is the science/technique behind this?
Science for this is simple: Hypothermia is not instant, and has a well known progression. Survival times in fresh ice water are 15 to 45 minutes; depending on your body build, subcutaneous fat, and muscle mass.
At 70 F (20 C) air temps a lightly dressed person loses heat about as fast as he makes it -- around 75 watts. If you are naked, you want it somewhat warmer than that.
Swimming pools are generally kept in the high 80s, and on stepping into them feel cool. Only because we are active can we tolerate the increased cooling.
"Increased?" Water conducts heat several hundred times as fast as air. So even 87 degree pool water causes you to cool faster than 70 degree air.
As the water temperature decreases the cooling rate increases.
Your body has several reactions to this to conserve heat: Surface capillaries contract -- skin goes white. Blood circulation to the limbs decreases. Body is trying to keep the core warm. You start to shiver -- initially this is voluntary: You can stop with an effort. Shivering basically exercises the muscles generating heat. As you get colder shivering becomes involuntary. At this point you still have some function. You can grab things with difficulty. Fine motor control is gone. You won't be able to tie a knot, say. As muscles get cold they become ineffective and stiff. You can no longer help yourself.
There are mental changes too. Outdoor people refer to the 'stumbles, fumbles and mumbles' Stumble mark the loss of accurate automatic actions. Your foot doesn't quite go where you told it. Fumbles mark the loss of intentional motor control. This can just be cold fingers having loss of sensation. It can also be cold muscles in your forearm which control fingers. Mumbles mark a cool brain that is no longer thinking coherently. This can also be cold cheeks. If the words that come make sense, it's cold cheeks. If they are irrational it's cold head.
When the brain stops functioning normally, the deficit follows the same sequence as getting drunk: Loss of judgement, rationality, increasing loss of motor control, passing out.
So going for a polar bear dip has very little risk involved. In a few minutes exposure to cold water you may end up shivering but it takes a longer soak than that to be in any danger.
Our outdoor program rules required that we do a dumping practice at the start of every canoe trip.
On several occasions this meant dumping in a bay, with solid ice pack not far from shore, and bergy bits floating in the bay.
We built a large fire on shore, and did our dumping practice nearby. Most of the guys were rescued into a canoe is under two minutes, and on shore within 5. 10 minutes later they were back out playing the role of a rescuer.
Edit: Rereading question you mention high altitude. This may affect building fires if this is in a subalpine ecozone. Firewood is often scarce and takes a very long time to renew. Under these circumstances having a towel to quickly dry off, and a sleeping bag handy may be the way to go.