Congratulations! You've completed Golden Sun: The Lost Age! There isn't really much after the Doom Dragon battle, except for a brief playable period with Felix and Issac where you can talk to the various townspeople of Prox who will thank you for what you've done. Oh, and quite a few scenes in the aerie of Mars Lighthouse, Prox and with Alex, especially where he went after Jupiter Lighthouse.
Once the ending's finished, you can choose where to save your clear game data. This is mainly for use in the Battle section of the game, accessible from the main menu. Also, with clear game data you can enter Easy or Hard mode of Golden Sun: The Lost Age.
Easy mode is where all your characters are at their old levels and stats (although they lose all their equipment). Basically this makes every enemy and boss a pushover. Hard mode is the opposite - all the enemies now do twice as much damage and have a lot more HP. If you really, really want to challenge yourself, and this really isn't worth it, you can play through the game again in Hard mode. Update along the way, and try beating Dullahan and the Doom Dragon at this difficulty. This time, Dullahan has 16,383 HP, while the Doom Dragon's 3 forms assume 7,500 HP, 6,300 HP, and 6,000 HP for a combined total of 19,800 HP.