After finishing the game on any difficulty, on any ending, you will end up being able to do a New Game+. This is not a full-fledged do-when-you-want option; if you want to do it, you have to do it after the credits roll and save a file.
Carried over automatically are the Demon Compendium, demon analyses, demon fusion allowances, play log, DDS data, and Apps of the type "Skill Expansion" or "Demon Skill+".
You can also choose to do a "Reincarnation Restart" , where you carry over stats, App Points, money, items, and equipment, or a "New Life Start" , where you start from scratch.
Either way, you also unlock the Master Difficulty.
So, I guess it goes without saying... Congratulations on beating Shin Megami Tensei IV!