Popman's hat is the hat that both Bossun and the fictional hero, Popman, wear. It has two horns on the top.
The concept of Popman's hat originate in Hime's origin story (see episodes 36 and 37). Popman is a fictional hero in a cartoon who Bossun and Hime admired. After Hime gives away the hat to Bossun, he begins wearing it with the Pop goggles. It is only until Bossun saves Hime and Chiaki from some thugs. Hime sees Bossun as her hero who saved her, the Onihime. She tells Bossun to always wear it, and Chiaki suggests Bossun to use it as a trademark for his club.