This one will take some time to compile, but I think it will be worth it in the end. During the Golden and Silver Ages, DC's writers and artists often created "other" Batmen. Some were fakes and some were inspired by the Caped Crusader, but they all wore the mask.
The first inspired version of Batman to my knowledge lived in the year 3000. His name was Brane, and the Earth he lived on had been conquered by the inhabitants of the planet Saturn. He and his young pal Ricky come across a time capsule from the 1939 New York World's Fair. Inside is film footage of Batman and Robin in action. This inspires them:
Another inspired Batman is the superhero of Zur En Arrh, who appeared in Batman #113:
One of my all-time favorite Batman stories is The Second Batman and Robin Team from Detective #220. Roger Bacon, a medieval scientist has discovered manuscripts documenting three of Batman and Robin's trips to the past (via Professor Nichols). He reasons that they must be time travelers from the future (correct) and that all people in that future time must dress similarly (oops!). He sends two young lads from his time into the future to learn the answer to his great question, which we learn here:
Only two issues later, we learned of the Brotherhood of Batmen:
Batman #62 included the story, The Batman of England. An English lord and his nephew decide to imitate our heroes. However, the Knight and the Squire dress differently (in plate armor) and ride motorcycles, ludicrously carrying lances as well:
Another future Batman appeared in Batman #105. Batman and Robin are surprised and concerned when a weird missile with a bat emblem shows up in the Batcave. It turns out the missile is capable of transporting them through solid matter, making their jobs a lot easier. The mystery is explained at the end:
Some others to be added: There's a three-issue series from the 1970s where Captain Stingaree is convinced that Batman is really triplets, and I also remember a rather silly GA story where there's an ersatz Dynamic Duo with Robin actually having a bald spot. Any others?