Purple Chocobos are not a mix of Blue and Red Chocobos, like might have been in FFVII, but instead a whole other type. They act as Synergists in your party, and are found in Bresha Ruins 300AF. Their stats are as follows:

HP: 25000

Chain resistance: 70

Stagger point: 300.0%

Common drop: Chocobo Plume

Rare drop: Afro and Purple Chick

Status immunities: slow, poison, pain, fog

Fire: Normal

Ice: Normal

Lightning: Normal

Wind: Normal

Mele: Halved

Magic: Halved

Notes: May flee from battle, capable of restoring health and casting Protect and Shell.

As a pet, their racial attribute is Critical: Protect, and they are a level one critter.