Not trying to nitpick, but I am talking from the perspective of traditional mountaineering rather than modern "action" sports.

First, there is a big difference between an alpenstock and an ice axe. An alpenstock is basically a long wooden staff with a steel spike. It was used originally by shepherds, hunters and mountain men as a walking aid much like a hiking pole today.

An ice axe is a different and much more modern tool, however. In the Alpine tradition it is always used together with crampons, which are considered *the* primary bit of kit for winter & glacier climbing. An ice axe is used primarily as a climbing aid (pick) and can be handy for cutting steps (adze). It can be used as a hiking pole of sorts, but it's generally too short and much heavier than a proper hiking staff or pole. Also, it's a lot more dangerous because it's easy to fall and impale yourself on the sharp end unless you're careful. So why bother if a hiking pole is all you need?

Obviously, everyone is free to pick their own gear and if an ice axe works for you, it's all good. However, oftentimes people pick some sort of gear more because it looks cool than for any practical value.

So from that point of view I would repeat what I have been taught since an early age, growing up in the Alps: don't bother with an ice axe unless you really have to. If you have the skills to use an ice axe properly you will know already when to bring it along.

As a basic rule of thumb, for any regular walking activity hiking poles will do the trick. If you need to negotiate some minor icy slope or the like, add crampons.

Whenever you need to do serious winter climbing however, using hands as much as legs, it's time to lay the hiking pole(s) aside and use crampons and ice axe.

YMMV.