It works likes DoS, AoS, etc. Once you beat the game you unlock the new game icon. If you want to start a New Game+, you just choose it and it'll save over your current save. You start over but with most of your stuff that you had from the previous game.
You also now have the choice to play in Normal (what you played before beating the game) or Hard Mode. If you choose Hard Mode, you have to choose from a level cap of either 50, 25 or 1 meaning the highest level you can ever get in Hard Mode is 50 (no 99 sadly).
Also when beating the game for the first time, you unlock 2 more modes, 2 more boss rush rush courses and sound test.
It has to be the good ending though, not the bad ending.
Oh, and welcome to the forums.
