Mountain hiking demands great gear and more education than your typical trail hike. Why? Because as you ascend breathing becomes less efficient, passage grows rugged, and rescue becomes more challenging. So consider your mountain equipment to be your life support system.
Where in day hiking you may be able to get away with wearing light-weight boots or shoes, in remote mountain hiking it's a good idea to wear heavier supportive boots to protect your ankles and feet. Walking with a sprained ankle on a day hike is inconvenient compared to trying to limp out of the back-country mountains with the same problem and a heavy pack. Thinking ahead and being prepared with the right mountain gear, guides, and information can literally mean the difference between a minor mishap and at times death. Take your safety seriously.
As elevation increases the atmospheric pressure decreases (amount of oxygen doesn't actually drop until over 50,000 feet) which means there's less pressure to 'push' the air into and fill your lungs. Because there is less air going into your lungs you'll fatigue more quickly. How much does the pressure really change? According to an academiclibrary.com article (that can be applied to mountain hiking) the barometric pressure can drop by 40% between sea level (average 760 millimeters of mercury) and 12,000 feet (483 millimeters of mercury). Obviously having 40% less oxygen available to your body (because of the pressure drop) will impact you.
Now if you had a 40% decrease in your ability to deliver oxygen to your body during an average day wouldn't you be concerned? Add in the fact you're asking more of your body in terms of keeping you warm in a cooler/cold environment, you are constantly losing water from your system, your reactions are slowed, and your not around the corner from the nearest hospital. Your mountain hiking equipment becomes a more important consideration the higher the altitude and the more remote your location.
Depending on how cold the temperatures you might encounter are you'll probably want some good insulated hiking boots (depending on the altitude and temperature you may need mountaineering boots), hiking clothes, sleeping bags, camping tents, and other equipment used for hiking rated for lower temperature than what you think you'll actually experience. This will give you an extra margin of safety. Also keep in mind that mildly uncomfortable mountain hiking gear will become a bigger problem.