Ak-Baba is Hilda's demon bird. Most of the time it is seen when she is riding it. It was also a "patient" to Oga and Beelzebub due to one of Beelzebub's toys. In episode 49 Oga rides on the back of Ak-baba.
Ak-Baba's body is mostly dark green though it has dark, but not navy, blue wings and a blue headress of the same shade. Ak-Baba has a gold-colored necklace studded all around with green stones. It is unclear whether they are precious or not. Hilda's Ak-Baba also wears dull yellow-colored bands around its hind legs' hips. It also has a silver armor on its forehead. It is also unclear whether the necklace and bands are necessary for survival/battle.
Ak-Baba is owned by Hilda and is obedient to her commands. It should be noted that in the anime, Ak-Baba, along with Alaindelon, refused to tell Hilda what Oga and his gang were doing when they were all actually planning a birthday gift for Hilda (whose birthday was not on that day), hence implying that Ak-Baba has a consciousness of its own.
It can be assumed that Alaindelon and Ak-Baba are on amicable terms as both owe allegiances to Hilda.