Here is the full list of all 14 Izanami's Dream Battle achievements.
Defeated Fuego for the first time.
Defeated Nieve for the first time.
Defeated Conejita for the first time.
Defeated Luna for the first time.
Defeated Izanami for the first time.
Defeated Punkin for the first time - All Clear!
Cleared the game on 'Normal' difficulty.
Cleared the game on 'Hard' difficulty.
Cleared the game on 'Why' difficulty.
Cleared IDB on 'WTF!?!?' difficulty.
Clear the game as Eione.
Clear the game as Florine.
Clear the game as Lauressa.
Clear the game as Punkin.