A secret boss can be encountered after the ending of the final boss cut scenes in Arcade Mode as long as specific conditions are met.

Boss	How to unlock
Secret Boss Mu-12 Encounter	Complete Arcade Mode with any character, excluding Noel, without being knocked out and with three or more Astral Finishes.
Secret Boss Ragna Encounter	Using Mu-12, beat Arcade Mode without being knocked out and with three or more Astral Finishes.