There probably is. Given the popularity of both Darkseid and Batman in the DC universe, it's a common and rather dark theme for fanfiction writers to explore. Some might write it to show the extreme power of Darkseid, while others may use it as a starting point for a more complex alternate universe story where Batman's death has far - reaching consequences for other characters.
I don't think it's a common concept. Most Batman fans love the character for his moral code which includes not killing. Fanfiction usually tries to expand on the existing lore in a way that is somewhat in line with the general perception of the character. So, stories where Batman kills are more of an exception rather than the norm in the fan community.
Yes, there are some out there. You can start by looking on popular fanfiction websites like Archive of Our Own or FanFiction.net. Just search for the specific tag'superman kills batman' and you'll find a variety of stories. However, the quality can vary greatly from one to another.
In a 'batman kills fanfic', it's a deviation from the norm. Fanfics often play with established characters in new ways. Here, Batman killing might be used to create a shock factor or to tell a story that challenges the status quo. It could be set in an alternate universe where the rules are different. It might also explore the psychological impact on Batman himself, as killing would go against everything he stands for in the regular DC Comics world. It's a creative exploration by fans who want to see what would happen if Batman took a more lethal approach.
I'm not sure there are any extremely popular 'batman kills fanfic' in the mainstream sense. Since it goes against Batman's core character trait of not killing, it might not have as wide an appeal as more canon - compliant fanfics. However, in some niche corners of the fanfic community, there could be some that have a following among those who like exploring darker and more alternative versions of the character.
It could imply that in some Batman - related fan - made fictional works, there are scenarios where Batman kills, which might be a departure from the traditional non - killing Batman in the official canon.
Occasionally, in alternate storylines or specific comic series, Batman has been depicted killing villains. However, this goes against his traditional code of not taking a life in the main continuity.
One possible reason could be mind control. Maybe some villain has taken over Superman's mind and made him do it. Another reason might be a fundamental difference in their ideologies that has escalated to the point of no return.
One common theme is the corruption of power. Maybe Superman's power has gone to his head, and Batman sees the need to stop him at all costs. Another could be a mind - control scenario where Superman is being controlled by an outside force and Batman has no other option.
One common theme is chaos. With Batman gone, Gotham descends into utter chaos as the Joker runs amok. Another theme is revenge. Sometimes the story is about someone seeking revenge for Batman's death. Also, the exploration of the Joker's victory is a theme, showing how he relishes in finally killing his nemesis.