Patient on Unauthorized Leave

Sometimes characters end up in the hospital. The sane thing to do when you're a patient in listen to your doctor's orders, no matter how badass you might be... but, for whatever reason, some characters just check themselves out early. They might have a mission to complete, they might feel just fine, they might simply hate hospitals. Going through the formal release process is optional. Going through the surgery that the character was checked into the hospital for is also optional.

No matter what the effects might be on the character's health, they just don't want to stay.

Anime and Manga

 * A Certain Magical Index: No matter how many times Touma ends up in the hospital, he always checks himself out early. He knows he should stay, but he can't afford the hospital bill an extended stay would produce. Good thing he's under the care of the best doctor in the city when he is in the hospital.
 * Played With in Magical Girl Lyrical Nanoha A's, in that

Fan Works

 * In the Spice Girls fic Astral Journey: It's Complicated, Melanie does this upon learning she had been sectioned for suspicious behaviors relating to her eating habits. She opts though the ceiling, where there she thought no one is going to look for her. She attempts a second time later on but ends up badly hurt. Despite this, her failed attempts actually gets the others giggling at her stupidity... even Emma.

Literature

 * In Monster Hunter International Nemesis a character does this because he is alerted to the fact that the conspiracy that injured him in the first place wants him dead to get rid of any witnesses. Notably he does not willingly engage in action, focusing on staying alive.

Live-Action TV

 * In the South Korean TV series Strong Girl Bong-soon, crime boss Baek Tak leaves the hospital after waking up almost fully healed thanks to a home remedy provided by the title character's mother, and goes on a pilgrimage seeking to change the direction of his life. His right-hand-man Agari and another of his Mooks, in considerably worse shape, sneak out of the hospital after him, and only catch up with him at the temple which is his destination.

Web Comics
"SHODAN: You've been holding out on us at Storytime in the Officer's Mess, Doc. BUNNIGUS: The naked runners are the most insistent regarding patient privacy."
 * Girl Genius had Klaus leave his room in the Great Hospital and stroll around in a medical walker (which allowed to take hospital bed and drip with him) the moment he could. Doctor Sun Jen-djieh rules the hospital with iron hand (not literally, but he doesn't need to) and has rather forceful objections when his patients ignore doctor's orders.
 * Schlock Mercenary reveals [//www.schlockmercenary.com/2010-08-24 tracker mod] for a medical scanner in case the doctor needs to "track a panicked patient fleeing in nothing but a hospital gown".