If you google the various photos of Bradley Cooper in Nightmare Alley (Searchlight, 12.17), you'll notice that aside from a natty moustache he's otherwise clean-shaven. (Okay, maybe a little chin stubble here and there.). For a film set in the mid '40s, which is when William Lindsay Gresham's novel along with the original Tyrone Power film version were released, this is period appropriate.