Endings, Stage Select, God Mode:
Beat game once and save clear data to unlock the ''Credits'' option. Carry over the saved data to a new game and beat the game a second time (select a different artifact in Roman Dude's stage) and unlock ''Jump to Game'' option. Beat the game a third time to unlock ''Eternal Mode'' -- basically Invincibility. Eternal Mode is accessible from the Jump to Game option only.

Secret Ending:
Beat the game three times with each of the three paths, and you should see a special ending movie.
Remember use your save data when starting a new game.