I have always carried a staff of some sort while hiking, whether hand-carved from a saskatoon (serviceberry) bush or an old downhill ski pole picked up at a garage sale for a buck.
The amount of stability it adds in rough or slippery terrain, and the protection it gives to my trick knee, makes it indispensible. Surprisingly, it helps you slow down while packing a load, meaning you have more control over where you step and how hard you land.
I generally only use one pole, except while snowshoeing. I switch sides on a regular basis.
A few years ago, I acquired a collapsible 'trekking pole.' It's not as sturdy as a heavy-duty downhill ski pole, but I like the fact it can go short and be used like an alpenstock, just marking time until you need a little extra push to hop a boulder or puddle.
I would be worried about getting tangled up with two poles while crossing a strong stream. But from my limited experience in wading streams, I would say that one pole is certainly worth having.