Artsy Moon

The Moon is shown in a bizarre or altered way - either because of actual physical changes, or simple Artistic License. Subtrope of Weird Moon, related to The Man in the Moon, Deface of the Moon and Bad Moon Rising.

If the moon is different because you're looking at an Alien Sky, you're in the wrong trope.

Examples of Artsy Moon include:

Anime and Manga

Bleach: For no apparent reason, the moon shows up as a very unlikely crescent.
Soul Eater has a crescent moon with a maniacally grinning face and hooked nose. It also has an Artsy Sun. It turns out the moon isn't just drawn that way, that's literally how it's shaped. It's not even an object orbiting the planet outside the atmosphere, it's just floating above the Earth in the clouds.
In Cowboy Bebop a third of the Moon has blown up into a giant Asteroid Thicket ring around the Earth, essentially leaving it as physically crescent.
Gankutsuou: Somewhat justified by the fact that it is colonized, thus having its surface altered, the moon sports a creepy skull-like face.

Hancock
In Treasure Planet, what appears to be a crescent moon in the sky is actually a bustling spaceport shaped like a crescent.

Krazy Kat, the comic strip, tended to show crescent moons. With a 3d-like representation.

Kingdom Hearts
Majoras Mask
Persona 3
In Earth Eternal, Earth's moon has had chunks blown out if it, with a Sickly Green Glow coming from between the pieces.

One of the two moons of Vytal in RWBY is shattered into a dozen or more pieces.

Our Little Adventure: The moon has a face. Justified by the fact that it's actually a moon goddess.

Coraline: Towards the end of the movie, the other world's moon starts slowly being eclipsed by a button-shaped shadow.
One Casper the Friendly Ghost cartoon has Casper decide to visit the Man in the Moon. Being a ghost, he has no trouble flying to the Moon, but as he gets closer, the smiling cartoon face on the moon slowly fades into just being a bleak pattern of craters, and he arrives bitterly disappointed. Fortunately, the place turns out to be inhabited by little Lilliputian guys, so he makes some friends after all.