An ice axe is often switched from hand to hand as a climber switches direction on a slope. A standard leash can be troublesome. The leash must be uncinched from one hand, placed on the other and cinched again only to be repeated the next time the direction is reversed.
A standard leash is the most comfortable option when swinging the axe like an ice tool (the way we differentiate between ice climbing tools and mountaineering axes).
We have heard some people claim an ice axe should not have a leash. We reply; there are places that accidentally dropping a tool would be very bad.
Their are other options. A length of webbing can be tied into a circle that slings over a shoulder and under an arm and is then tied to an ice axe.
A length of rope can be clipped into a biner on the front of a harness. It must be long enough to reach arms length. The leash at this length gets in the way. We often use a leash of this length with bungee pushed through the middle of the webbing. The bungee is shorter and the webbing is bunched up over it. The leash will then stretch to the length needed and stay out of the way. An added benefit is that pulling the the webbing leading to the axe through the loop for the biner to clip into makes a good wrist loop when a harness is not needed.
We tend to use a standard leash for more technical terrain and the stretch leash the rest of the time.