Woo hoo! Chance for my 50's Batman fandom to be of use!
The costume Batman is currently wearing is similar to the costume worn by the Batman of Zur-En-Arrh. The Batman of Zur-En-Arrh and his planet appeared in one story, in Batman #113 entitled "Batman, The Superman of Planet X". The Batman of Zur-En-Arrh observed the heroics of Batman and decided to emulate him and become a Batman for his planet. The Batman of Zur-En-Arrh transported the Batman of Earth to his planet to help combat a robot invasion, as the elements of Earth make Earthlings super-human on Zur-En-Arrh. After defeating the robot invasion, Batman returns to Earth and wonders if what happened really occurred...
Except for Batman and Son, most of Grant Morrison's Batman run has brought back stories from the 50's Batman era into continuity. Zur-En-Arrh appeared as graffiti on walls in early issues of his run. It later showed up as the trigger word that knocked Bruce out and allowed for his capture by the Black Glove. The Bat Radia reappeared (although it may be a delusion on Bruce's part) when he stitched together the Zur-En-Arrh Batman costume.
As for Bat-Mite, during the 50's era he was an extradimensional imp who idolized Batman and came to Earth to "help" him in his crime fighting. More often than not, his attempts to assist Batman were more harm than good. Grant Morrison as brought him back into continuity with a much more sinister bent.
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