Considering that Conan went toe-to-toe with Captain America, he could easily go toe-to-toe with Bats. They both represent the pinnacle of human excellency in the physical domain and are also as smart and cunning as they come. I say it would boil down to endurance and willpower. I give the victory to Conan based soley on endurance.
Go suck a sword.
That is the most garbage idea I've ever heard. Bats is a master fighter in like 350 different languages. He doesn't go down against anyone physical style without a fight. Got that, punk??