Ironwood is a generic term used for alot of hard woods. Desert ironwood is a specific species. Hop Hornbeam is sometimes called ironwood. There are others.
Not all would make a good walking stick. Desert ironwood is extremely dense. It sinks in water. It's also very decorative, beautiful colours and grain, especially in burls. It'd probably be too brittle to make a good striking implement with. I know that my knives only survive about 30 strokes on the stuff before being dull and burred up. It's brutally hard. About 3x as hard as rock maple in my experience. Think carving aluminum with a knife and you're close.
Hop Hornbeam, on the other hand, is a better choice. Still quite dense, but this stuff can float. Makes a slim stick able to deliver a real blow. I've only got limited experience with this stuff, but it's hard.
In the end, if you want a stick that will survive a whack, study up on the most manufactured and used striking weapons on earth. Baseball bats, hockey sticks, lacross sticks, etc. Their choice of wood is telling.
Hickory, Maple, Ash. Maybe some oak.
This wheel is long since invented.
J