In reply to Inhambane:
If I were winter walking in the Alps I'd be thinking snowshoes, snowshoes, and walking poles. Even at low altitude (900m) you could be looking at thigh-deep snow at new year. Without snowshoes & poles this is essentially impossible to deal with, with them it's a delight.
Aside from that, you need to have an attitude of self-sufficiency. There will be plenty of people by ski infrastructure, but walk along a path for 10 minutes and you're on your own - once the mountain huts close the mountains really empty out. You will potentially see skitourers if there's somewhere for them to ski down on the route, but walking paths without off-piste skiing, winter mountaineering peaks or open huts will be empty. So, I'm in hiking boots and carry /wear snowshoes - they all have crampons built in so you shouldn't have trouble with ice unless you're on steep scrambly ground above the treeline where the size of the snowshoes becomes a pain. Clothing is no different to UK walking, but definitely take a warm jumper/fleece/duvet jacket to wear when you stop. Full waterproofs.Hat and shelled warm gloves - wooly gloves are no good when they get wet with snow. Simple First aid kit. Phone / GPS - many paths above the treeline rely on painted waymarks which can be obscured by snow. It's not necessarily a problem to follow your nose, but get a whiteout and you'll find the backup of GPS useful for avoiding epics. Flask of hot drink is useful to give you a warm mug to thaw your fingers out on, as is a decent amount of food. Leave a routeplan and estimated finish time with someone so they can initiate a rescue when you don't make contact at the end of the day.
I've taken up snowshoeing in the Bavarian alps as I have relatives near there and I love it - there's many nice day routes around the top of the treeline so you get the 'winter wonderland' amongst the trees then the views from the tops without the worry of avalanche risk or other mountain problems. I like to find somewhere with a hut and lift at the end of the route - I can enjoy a beer and good hut food then avoid the downhill slog to the valley with the lift or better still, rent a toboggan and slide down.
Post edited at 12:12