okay, I have seen theories about what sort of relationship batman and the joker has. and...one of them, I really disagree with is the one, that they are gay for each other. eh...bullcrap. Im sorry, but no. I have nothing against homosexuals, but neither the joker or batman are homosexuals. yeah, there are homosexual characters in DC comics, like midnighter and batwoman, but not the joker and batman. anyway, I think I should explain a bit how the relationship between batman and the joker is. (BTW, Im not saying Im right, just saying how I see it, after reading lots of comics, seen TV Shows and whatnot.)
first of. its true that they are both obsessed with each others so to speak. not in a gay way like I said, but let me explain.
Batman is obsessed with the joker, because the joker is the ultimate monster he has faced. the joker kills with every breath. he can make anything into a deadly weapon. (he has literally killed a person with one nut) so, whenever The joker escapes, Batman basically puts the likes of penguin, two face and black mask aside, cause with the joker the body count will get larger, and he has to catch the joker as soon as possible, because if he don't, then more people will most likely die. so, yeah. its basically have to make sure no people will die because of him, is the reason of batman's sort of obsession. but that obsession doesn't always have to be the joker. for example the joker has escaped but now Darkseid is invading with he's army of parademons. then batman will put the joker aside and completely focus on darkseid, with the aid of the justice league. because he knows that even more people will most likely die with Darkseid. but another reason for batman's obsession with the joker, is that he also don't know who the joker is, or anything from before joker became the man he is today. he is a detective and can explain most things, but the joker? That is the one man he can't explain. its sort of with the riddler's obsession with batman so to speak. who is he? And why does he do what he does? So, my guess is that batman thinks that if he can find out who the joker was before, and why he became what he is now, maybe he could find a way to end the killings permanently. so all in all batman's obsession comes from the purpose to make sure nobody dies.
then we have the joker's obsession with batman. and he's obsession has actually changed over time. but it all comes from the fact that batman was sort of responsible for the joker falling into a vat of chemicals. but back when the joker first appearance in gotham, he killed off some of the co owners of the ace chemical plant. and back then he basically wanted to kill batman, so at the beginning it was basically revenge for making me what I am. but as they continued battling over time, the joker ends up actually enjoying to fight batman. like it gives him a thrill. same thing with murdering at first it was basically about revenge, but he ends up enjoying that too. and this is where the whole "You complete me" thing comes from. like for the joker fighting batman is a drug and he needs it. you see when the joker fights batman, batman always comes back, and does the same thing. the joker is sure, that deep down the world is just like him and that all it would take would be one bad day. but no matter what that doesn't apply to batman. Joker has given him hell, and yet batman always fights back, never kills and refuse to become like the joker. "You truly are incorruptible aren't you?" batman is different from all the others. so the whole fighting batman becomes a game to the joker. one he enjoys! then we come to what I can say the next part of joker's obsession. remember when I told you the joker knows he is a fictional character? Well, he also realizes that he is the villain of the story and batman is the hero. and in he's twisted mind he is like "Since batman is a hero, I'll help him becoming the strongest hero there is!" so he continues challenging batman with stuff to make batman stronger. thats the reason for "Death of the family" were the joker thinks batman's family has made him weak, and he plans to get rid to them to make batman strong again! to become the true bat-king of gotham.
so thats basically how the joker's relationship to batman has evolved. And as you can see. NOT GAY!
the only other relationship the joker has to batman I need to explain is the one from The dark knight returns. people use this as an example for joker being gay, since the joker says "Batman...Darling." and once again. ITS NOT IN A GAY WAY! As before batman completes him! He enjoys fighting batman! and when batman vanished the joker didn't see a reason to continue anymore. the joker basically becomes a vegetable after that. without batman, crime has no punchline. but when batman returns, the joker sees the thrill again! when he says Darling, he doesn't refer to batman as he's lover. he refers to the thrill of the fight! he refers to the fun itself he has had fighting and torturing batman! So, yeah. once again its not gay!
well hope that explains a bit. and yeah....stop saying that batman and the joker are gay for each other, cause they are not.