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.
Originally posted by doctorstrongbad
Conan takes out Batman. They both rush each other batman goes for a jump kick and conan gacks him with his sword. The end