Monster companion

Monster companions first appeared during the 2nd event. Although this event marked their first appearance, it wouldn't be until the 7th layer was released that they would become commonly available to players.

When a player acquires a monster companion, they gain an item called the Bridge of Bonds. When the 7th layer was released, the Bridge of bonds gained the restriction of only having 1 equipped at any given time.

Mythical Beast
As of now, we only know of the existence of 4 mythical beast eggs. They were items that could be acquired during the 2nd event. The bosses that guarded the eggs were believed to be far outside the range players could deal with. There was also no clues or machines around that would lead to the eggs hatching. Because of this, the developers believed it would be impossible for players to acquire and hatch the eggs.

Silverwing was the guard of the fox and turtle mythical beast eggs before he was defeated by Maple and Sally. Maple chose the egg with a deep green color to it and Sally took the reamining egg with a faint violent color to it. Maple's egg later hatched into a green turtle she named Syrup and Sally's egg later hatched into a white fox (with a shimenawa) she named Oboro. Though Oboro himself is white, his fox fire is violet.

The primary Sea Emperor was the guard of the bird and wolf mythical beast eggs.

Rare
Bear: The first known Rare-tier monster companion is the Bear that the twins, Mai and Yui, acquired. The bear can be acquired by following it to it's cave and defeating it in combat. The bear probably has the Enlarge skill because just before the fight begins, it grows large. An unknown difficulty was skipped because Mai and Yui were able to one-shot the bear.

Other
..more later as common monster companions make their appearance..

Customization
With the release of the 7th layer and the ability for all players to freely work towards acquiring a monster companion, the ability to customize their appearance as well as skills/effects became available.

Shortly after the twins acquired their bears, Iz asked what they planned to name them and how they want to customize them. Mai chose to name her bear Tsukimi (ツキミ), make his fur to a glossy black, and make his skills/effects glow green. Yui chose to name her bear Yukimi (ユキミ), make his fur white like a polar bear, and make his skills/effects glow pink.

Known Companions

 * Syrup (Turtle Mythical beast) ♥ Maple
 * Oboro (Fox Mythical beast) ♥ Sally
 * Tsukimi (Bear Rare beast) ♥ Mai (glossy black fur, green skills/effects)
 * Yukimi (Bear Rare beast) ♥ Yui (white fur, pink skills/effects)

Trivia

 * Though there is no source; Syrup is likely a reference to the Spirit Turtle as it represents longevity and Oboro is likely a reference to the Fox Spirit as it represents mischief. They match up to how Kaede and Risa play NWO.
 * It may also be in reference to the Four Beast Gods of Japanese and Chinese Myth, namely Genbu, Suzaku, Seiryu and Byakko, represented by the Turtle/Snake, Pheonix, Dragon and White Tiger.