Successfully complete the game. You can start a new game session with all previously obtained stats, items, and money.
Defeat the Almighty and save the game. Load the saved game, and go to where the Bosses were located earlier. Open the black crystal to the Bosses again.
Defeat the Almighty and complete the game and save it. Do not start a new game. Instead, load the saved game. Access and complete the new map and defeat the Almighty at the end. You can now select Ayuta and Mana as well as Olgar and Vhati by using the arrows at the bottom of the character selection screen.
Fight against Piu Piu
Defeat the Qualia with each pair of fighters. Save the game. Load the saved game and talk to Piu Piu. Answer "Yes" to his question to fight against him.