Barbatos

Barbatos or the Hyper-Adapter is a bat-like cosmic being which is attributed to being the in-universe inspiration

behind the  identity. Barbatos is typically percieved as being a bat-like demon, nightmare or a literal bat affiliated with, the Wayne Family and Bruce Wayne.

Prime Earth
In Prime Earth continuity, the Hyper-Adapter was a being kept by the evil New God, Darkseid inside of an item called the Ancestor Box. Darkseid had been mortally wounded by Batman and in-turn sent him back in time with his Omega powers. He proceeded to release the Hyper-Adapter which could freely traverse time to pursue Batman throughout all of existence, gain his form, and destroy him.

Several stories written prior to the creation of the Hyper-Adaptor feature beings which can be attributed to it. makes referance to a bat-demon which was said to have once plagued Gotham in the city's folklore. 's seemingly random nightmares of a bat terrorizing him which he had throughout his childhood in  could have been Barbatos.

New 52
In the New 52, Barbatos was reimagined as being the god of the Dark Multiverse and objectively, the first incarnation of Batman in the entirety of the New 52's multiverse. Barbatos was spawn of the cosmic god known as the World Forger in-order to reign over the Dark Multiverse. The Dark Multiverse was a plane of existence populated by realities born from the nightmares, fears, and darkest desires of the beings of the central multiverse. Barbatos came to murder the World Forger and aimed at taking over the rest of the multiverse.

Wishing to have a portal to the normal Multiverse, Barbatos forged the Batman to be in his image. When Bruce Wayne was starting out in his career as a vigilante, Barbatos took the form of a bat in Wayne Manor which Bruce took as having been a sign from his late-father to take up the cowl. He also became worshiped by a group known as the Cult of Judas which over the eras evolved into the .

Centuries later, the Court began preparing Batman to serve as Barbatos's portal to the multiverse while Barbatos himself began to assemble his own league of evil Batmen created in the Dark Multiverse called the Dark Knights;  the likes of which included the Batman who Laughs.