The stippled decoration on the axe face, I agree with Runjeet, does seem to correspond to this kind of decoration in India in later period, 19th century. It should be noted that it is also a characteristic of markings on the weapons in Bikaner armoury in Rajasthan, which simply reveals that the punched dot style for decoration and inscription was popular in these regions....I am not suggesting these are from Bikaner, nor armoury applied, only the style of decoration.
I recall seeing the small head, long shaft type utilitarian walking axes that Tim has mentioned for Eastern European regions, as well as similar as mentioned by Ibrahim on the Arabian peninsula.
While serving as walking staffs these certainly would double as weapons much in the sense of the well known sword canes of Europe....so you're right Gene, I think you got a weapon here
I think shorter shafts would simply indicate more of a belt axe.
All best regards,
Jim