Finish the game with any character. When you start a new game, you'll retain all stats, items, and money.

Beat the Almighty and save your game load it back up and where the bosses were before there will be a black crystal to open and fight the boss again

Beat the game once. DO NOT START A NEW GAME. Just load the old one again and a new map will be open go through it and defeat the Almighty and save at the end. When you turn it back on the character select screen will have arrows at the bottom to switch between the characters.

Defeat Piu Piu with all characters to unlock a new title screen.

Defeat the Qualia with each pair of fighters then save your game. Load any one of your saves and talk to Piu Piu. Answer yes to his question to engage in battle with the legendary scientist.