The weekly meeting of the
Shattered Hand Clan is this way.
All joking aside, if you can wrangle some interesting rules about it, it should be a cool concept to play with. The stump would need some sort of removable attachment system for the blade (or maybe just a sheath) so you can avoid poking things by accident while you're sleeping, but besides the usual issues with having only one hand it's pretty workable.
Although instead of an axe head, I'd suggest a stabby blade with slashing potential, like a long push dagger or a
katar. It's generally a lot easier to punch than to chop, and since it doesn't have a handle and it's attached to the forearm, you wouldn't have much leverage for slashing-hacking with in the first place. Depending on how the stump was capped (if the bones in the forearm can still rotate), you could even use it as a knife for more delicate operations. Just make sure you wash your stab-hand thoroughly before you use it to cut food while in camp! You don't want any bits of zombie touching the steak, or the cleric will have to spend a while casting remove disease spells on the party!
Great icebreaker at parties, too!