If a creature changes player who controls an enchantment?
Same thing (though less likely to move.)
It's worth noting that some creature Enchantments (Auras) specify "enchant creature you control." If control of that creature changes to a different player, then that specific enchantment would "fall off" and go to the graveyard as a state-based action. (For example: Spirit Loop)
Additionally, as the original controller of the enchantment retains control of it even if the creature changes controllers, then the original controller is the one who gets to use certain abilities some auras have. Abilities to return the aura to hand (Flickering Ward, Agoraphobia, etc.), or to untap the enchanted creature (Crab Umbra) would still be controlled by the person who cast the enchantment, but other auras that grant abilities specifically to the creature (like Madcap Skills, which says "enchanted creature gets +3/+0 and menace, or Lightning Prowess, which gives a creature haste and a tap ability) are used by the creature's new controller.