by fighter fish, do you mean Betta? I think they sleep; or at least rest - in fact, I've read they like a leaf or something soft to rest on. I've only had mine a few days and I have found him resting occasionally on the plant I have in the tank for him.
My O's definitely sleep or rest - I find them laying side by side resting on the bottom - one kinda leans over and the other is usually more upright right next to the other one
here is a bit more info about sleeping oscars.....
www.oscarfishlover.com/sleeping-oscar-fish