First, it's important to get the nomenclature straight.
An "ice axe" usually means the type of axe used for glacier walking and two-handed self-arrest. This has a smooth, straight shaft for easy plunging into snow as a crevasse probe or temporary anchor, enough length to be used as a cane, a nice big adze for digging, a comfortable head for gripping, and a pick that's designed chiefly for self-arrest. The BD Raven is a good example of an "axe".
An "ice tool" is designed to be used in one hand for vertical ice climbing. It has a short, curved shaft with a grippy handle, a narrow pick with a fancy curve that is stable when inserted yet easy to remove from the ice (but isn't very good for self-arrest), and is usually used in pairs - one with hammer, one with adze, and the adze is often reduced in size. For an extreme example of a "tool", look at the Grivel Monster.
There are lots of "hybrid" designs, but they are all compromises - not ideal for either task.
What do I want in a design? Lighter and cheaper! Also, there is room for improvement in leash systems. If you're not up for inventing the perfect leash (adjustable length, quick and reliable hand insertion and release, some way of ensuring it won't dangle or tangle), make it easy for people to attach their own leash (or none).
For glacier axes, most people wrap some kind of insulation around the head so they don't lose heat through their hand in cane position. That insulation could be built-in.