Each Abyssea Zone has a Boss. Beating them has rewards.

When you've beaten the 1st and 7th of the 9 you receive a Lunar Abyssite, totalling up to two, and are able to infuse another Atma of your choosing.

Abyssea - Altepa - Bennu

Abyssea - Attohwa - Itzpapalotl

Abyssea - Grauberg - Amphitrite

Abyssea - Konschtat - Kukulkan

Abyssea - La Theine - Briareus

Abyssea - Misareaux - Cirein-croin

Abyssea - Tahrongi - Glavoid

Abyssea - Uleguerand - Resheph

Abyssea - Vunkerl - Sedna