A: While one of the most obvious ways to get rid of a Creeper attached to one of your Keepers is to trash or move them both, there are some versions which include cards which let you detach the Creeper before you get rid of it.

Cards that let you separate the two:
In Star Fluxx, and the various Star Trek Fluxxen, the Engineer (or version-specific named Engineers) can fix a Malfunctioning piece of equipment.
In Star Fluxx, the Doctor can cure Brain Parasites.
In Cthulhu Fluxx, the Sanitarium can cure Nightmares or Insanity, and the Dreamer can cure Nightmares.
In Anatomy Fluxx, It’s Contagious! allows all players to detach and move Bacteria or Viruses to another player, while It’s Cured! causes all Creepers to be discarded, detaching from Keepers first, if they are attached.
The Action: Creeper Reassignment, found in all Trek Fluxxen and Cat Fluxx, lets you move a Creeper, detaching it first if it is attached.

Anything else, unless it specifically says you can detach the Creeper, is going to affect both cards if they are attached.