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.
Well, perhaps Aang realizes that Katara has feelings for someone else and decides to let her go for her own happiness. This could lead to a new phase in their friendship where there's less tension and more mutual understanding.
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.
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 cases, the anger could be a catalyst for a more romantic development. Aang expressing his strong emotions might make Katara realize the depth of his feelings for her. Or it could be the other way around, Katara's reaction to his anger could make Aang see her in a new light, leading to a shift in their relationship dynamic towards a more romantic one.
They could embark on a quest to find a lost relic. Aang's airbending skills and Katara's waterbending would be crucial in overcoming the obstacles like ancient traps and dangerous creatures. Their teamwork would strengthen their bond.
One great fanfiction idea could be them sharing a picnic by a beautiful waterfall. They could talk about their dreams and how far they've come together. This would be a really sweet and innocent moment that shows their deep connection.
You should check out 'Aang and Katara: Love in the Four Nations'. It takes them on a journey across all the four nations, and during this journey, their relationship blossoms. It also gives great insights into the different cultures of the Avatar world and how Aang and Katara interact with them.
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.