Chrono Trigger SNES Rom question.?

Warning, this is a pretty specific and nerdy video game question. I just finished defeating Lavos in the game, Chrono Trigger. I have been following a walkthrough for the game and it recommended that I defeat the Black Omen to get the "New Game+" option. This means that I can play through the game from the beginning with all the goodies from the first time through. Ok, I did all of this and defeated the game and watched throught the ending of the game. "The End" came up and then nothing. How do I get the "New Game+"? Is it not possible with the Rom game played on the computer. Does it only work when I am playing it on the Super Nintendo? Thanks. I warned you, this is pretty nerdy.