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:
Chain resistance: 70
Stagger point: 300.0%
Common drop: Chocobo Plume
Rare drop: Afro and Purple Chick
Status immunities: slow, poison, pain, fog
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.