Any recommendations about the boots?
Or what to check for?
Your main concern is to stay dry and minimize snow incursion. Either wear ski pants that act as gaiters, use gaiters or use boots and socks that are high enough that you won't get snow down the sides.
If you buy boots with insulation, try not to compress it. Otherwise be aware that if you don't keep moving, your boots will eventually cool to ambient and it's pretty hard to get that temperature back up.
Check grip? Hard to test but warm doesn't necessarily mean any good on slick ice. Spikes work well if you're going on a hike and there's a lot of packed snow mixed with ice.
Don't forget good socks. Doesn't need to be anything fancy, but wool is by far the best material (not necessarily merino as it tends to be too thin). You may need to size up because of the extra padding.
Also luxury, but fan assisted boot drying/warming stations are great. They make quite a big difference if you go out a lot because moisture build-up takes ages to dry otherwise.
Pick a size larger than you would usually do, unless they're explicitly designed as winter boots. In cold weather you'll want wool socks as well as regular socks and that requires some extra space.
When given a choice, soldiers will tend to choose something like Salomon Toundra.