Another Puzzle World
At the Main Menu press L, Up, R, Down.
A tiny character will appear in the lower right corner. Select Start Game now and when you select "Puzzle Mode" it will say Another World.

Level Skip
During game play press "Z", "R", "L" and "A" at the same time. This will bring you to the next stage.

Hidden Character Select
At "Puzzle Mode Select" screen press Left, Left, Up, Down, L, R, L, R, L+R. This will bring up a Character Select screen where you can choose between the two dino's and also a few bosses.