How to Complete Unplumbed Depths
The quest says that the objective is to get the treasure, but you can't actually open the chest until you defeat all foes. The enemies are some Bloody Orbs, a Bomb, some Golems, and a Lamashtu.
The Lamashtu is especially dangerous because of her Eternal Sleep ability, which targets all units (including enemies) and has a chance of inflicting Doom and Sleep on every unit. The Lamashtu herself will still be awake after using the move. You might want to equip someone with a White Staff, because if you hit a Doomed unit with the White Staff, Doom will be removed from that unit. You can also use the White Magick ability Refresh to remove Doom.
Equip Ribbons on as many units as you can.
The Golems can use Helmsplitter to destroy a unit's helm, so be sure not to equip any rare helms in this battle.