It could also be due to a misunderstanding. Maybe Aang does something that Katara misinterprets as a betrayal or a lack of respect for her values. This misunderstanding might lead her to make the decision to break Aang's heart and end their relationship.
Perhaps Katara falls in love with someone else. In a fanfiction scenario, she might meet a new character who she feels a stronger connection with. This new relationship could cause her to break things off with Aang, shattering his heart in the process.
Aang might go into a deep state of depression. He has always been an emotional character, and having his heart broken by Katara, someone he loves deeply, could send him into a downward spiral. He might isolate himself from his friends and stop practicing his bending for a while.
It could also be that Katara was being overly critical of Aang's progress in learning new skills. Aang might feel frustrated and angry if Katara is constantly pointing out his flaws instead of being more encouraging, especially if he's already trying his best.
It could be that the fanfiction writers wanted to explore a more complex and darker side of relationships. They might have written scenarios where Aang hates Katara due to jealousy, perhaps over her friendship with another character. This would add an interesting twist to the original relationship dynamics between them.
One possible reason could be their different approaches to life post - war. Aang is the Avatar and has a whole world to look after, while Katara has her own tribe and personal goals. Their busy schedules might pull them apart. For example, Aang might be constantly traveling to different nations for diplomatic and Avatar - related duties, leaving Katara feeling lonely. Also, they might have different ideas on how to handle future threats. Aang may prefer a more pacifist approach which Katara might not always agree with, especially if it endangers the people she loves.
In some fanfictions, it might be part of a plot to create tension. The writer wants to show that their relationship isn't all smooth sailing. So, Katara hurting Aang can be a device to make the story more interesting. It could also be related to different character arcs, like Katara trying to become more independent and in the process, her actions accidentally harm Aang.
Perhaps Aang has a vision of the future where Katara is happier without him. So, in the fanfiction, he decides to respect that future and let her go.
Maybe in the fanfiction, Aang leaves Katara because of a misunderstanding. For example, he could misinterpret something Katara said or did regarding their relationship. This misunderstanding could lead him to believe that it's better for both of them if he goes away. He might think that he is protecting Katara in some way by leaving.
The writer might be trying to add some drama to the story. By making Aang hate Katara, it creates a conflict that can drive the plot forward. For instance, their friends might have to work hard to resolve this hatred and restore their friendship.
In fanfiction, their married life could be filled with adventures. They might travel together to different parts of the world, using Aang's airbending skills and Katara's waterbending. They would also be involved in helping the people, just like they did before getting married. Their love would grow deeper as they face new challenges side by side.
Well, it could be a very emotional moment. Maybe in the fanfiction, they are in a beautiful, secluded place, like a hidden cove by the sea. Aang might be a bit nervous, but his love for Katara gives him courage. Katara, on the other hand, would be full of affection and warmth towards Aang. They would probably exchange some soft words, and then share a gentle first kiss, starting this new chapter of their relationship.