For many people, winter is the best time to be on the hills.
A beautiful sunny, freezing day on ice and snow covered ground high in the hills can be one of the best days of your life. It goes with out saying that winter days are colder, shorter, and can be wetter. Snow and ice adds an extra dimension.
When there is snow on the ground, an ice axe and crampons should be regarded as essential. You may not need them, but if you do, there is NO substitute.
Four season boots will keep your feet warm and dry, as well as provide a solid platform to fix the crampons. Put them on BEFORE you need them and take them off AFTER. Hopping on one foot on steep ground is not the time to try and put them on. Anti-balling plates, or a thin carrier bag fitted between boots and crampons will stop snow building up and freezing on to your feet!
Ice axes are a personal choice, but if you're walking don't be lured in to thinking a climbing axe will be better. The steepness of the pick and curvature of the shaft will make it much less useful on anything but very steep ground.
Carry the axe in your hand, or down your back between your rucksack straps where it is accessible as soon as you think you need it. It's NO use attached to the back of your rucksack.
Take buying advice from a reputable outdoor shop.