Another World
At the title screen, press L, Up, R, Down. If you entered the code correctly, a small green face will appear in the lower right corner of the screen. Select "Puzzle Mode", then choose the new "Another World" option. Alternatively, press A, Up, B, Down at the title screen.

Bonus characters
Select "Puzzle Mode". Then at the puzzle selection screen, press Left(2), Up, Down, L, R, L, R, L + R. Now Bub, Bob, and the Boss characters may be selected.

Play as Bub
At the game selection screen, press Z + L + R + A.

Level skip
While playing the game, press Z + L + R + A.