The best and most appropriate boot to wear when doing any mountain hiking in winter months is a mountaineering boot. A three season hiking boot just is not going to keep your feet warm enough and protected enough for long hours out in the mountains. For trekking through snow, you also want a stiffer sole that has plenty of insulation on the bottom and around your foot. A mountaineer boot will go farther up your leg than a regular hiking boot, which will do a better job of keeping snow out. Mountaineering boots will also allow you to attach crampons when you need to climb up icy areas.
Here are Light Weight Single mountaineering boots (appropriate for temps around freezing to 0 degrees):
When temperatures and conditions get extremely cold you'll want to use heavier double boots which have a outer shell and inner liner that can be taken out of the boot to dry out.
Here are some Double Plastic Boots that are used for mountaineering: