In reply to Dibyendu Das:
Do you want technical axes for ice or rock climbing? Or walking axes for mountaineering? Or something for training on indoor walls?
The others are right, modern axe designs are really quite refined, the materials fairly exotic but we are looking from a 2020 western consumer perspective, at their core they are simple tools, axes, which would until pretty recently have been hand forged by blacksmiths and mounted on wooden shafts.
To make a walking or basic climbing axe (totally reasonable) with basic tools I'd make a pick from the flattest portion of a leaf spring. Cut the profile with an angle grinder, flatten and thickness gradually with flap wheels. I'd probably braze that into a thin wall steel collet of seamless tube (bike frame), glue and bolt/rivet the assembly onto a wood or fiberglass tool handle carved/ground to shape. Bolt or braze the adze/hammer into a mechanical joint.
Time, grinder and cheap materials can yield something nice and functional.
Jk