I'm not so sure an ice axe is a good idea unless you have some training under your belt.

An ice axe is basically a specialized tool used in serious winter climbing and can save your life if you fall on an icy slope. But only if you're properly trained in self-arrest techniques.

Which is pretty technical stuff, not something a newcomer could learn from a book. From what I've been taught, one should never use an ice axe without crampons. The two go hand in hand, but again winter mountaineering and ice climbing is serious stuff and should not be attempted without qualified training.

An average hiker will never venture that far and will never need that kind of specialized gear. In unskilled hands, an ice axe is more trouble than it's worth. An ice axe is no substitute for a walking staff or pole, it's primary purpose is completely different. It might also inspire false confidence, which is a very bad thing on a steep icy slope unless you know exactly what you're doing.

If you do want to try hardcore winter mountaineering though, do learn from a qualified instructor first. It will also make it much easier to pick the right gear for the job to begin with.