Bonus Dungeon:
This dungeon has 50 floors and each floor is randomly generated based on the average levels of your party.

The 50 Floor Dungeon - Beat the game once and then fly around on the moon's surface until you see a boulder shaped like a head with a face.

The Cave of Trials in Final Fantasy IV Advance:
An eight floor dungeon containing weapons and armor for the party members you couldn't use for the final dungeon in the original. Once you unlock Party Switching and head back to Mysidia, you'll be informed of a new challenge at Mt. Ordeals. A scene will occur once you get there and the Cave of Trials will be accessable.

Music Player:
Beating the game once unlocks the Music Player. It can be accessed under "Extra" on the title screen.