For me, I cold shower for mental reasons more than physical reasons. I think it's a great way to force myself into doing something uncomfortable, plus train myself to handle discomfort and stress.

Although, 1 year into regular cold showering, my late Spring allergies to cottonwood trees seem to have completely vanished. It could be because of other things, but the most obvious and major change I had was continued cold showering for about a year. I also don't seem to get sick much, and even if I do get sick, I seem to recover rather quickly. So I guess there's probably some physical benefits to cold showering as well. Not trying to convince you to do it or anything, just reporting on my experience from consistently doing it.