Tried running the game on my 4K display. Even with the config set to 3840x2160, it refuses to boot at any higher than 1920x1080. Using a GTX 1070 on this one.
Topic on Talk:Batman: Arkham Asylum
OK, doing a complete uninstall/reinstall and wiping my config folder seems to have fixed it.
Try the BAA Advanced Launcher. It could have better Video settings for modern systems.