She could be a very determined person. Growing up with Naruto as her father, she would have heard all about his struggles and achievements. So, she would likely strive to be as great as him. Maybe she'd be a bit of a rebel too, since she's not growing up in the traditional Konoha environment.
In this fanfiction, Naruto's daughter could be a complex character. She could be both strong and vulnerable. On one hand, she has Naruto as her father, so she has the potential for great strength. But on the other hand, growing up away from Konoha means she may face some insecurities. She might be curious about the village her father left and the people there. And she could be someone who is constantly trying to find her place in the world, whether it's in the ninja world or in the relationship with her father.
In such a fanfiction, Naruto leaving Konoha could lead to his daughter having a different upbringing. She might be raised in a more secluded place, away from the hustle and bustle of the village. As a result, she could have a more independent personality. And she would probably have to face the challenges of not having the full support of the village that Naruto once had. But with Naruto as her father, she'd surely be strong - willed and determined to make her mark in the ninja world.
It could be something like Sakura, after Naruto's friend. It would be a way to show his respect and friendship for Sakura. Or maybe something related to fire, like Kaen, since Kurenai is associated with fire - based jutsu.
Well, Naruto might leave Konoha because of the political unrest within the village. He may feel that his ideals are not being supported by the current leadership. For example, if they are more focused on power and control rather than the well - being of everyone. And perhaps he's just tired of the never - ending cycle of conflict that always seems to involve Konoha, and he wants to find his own path away from it all.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
Perhaps he discovered some dark secrets about Konoha that made him lose faith in the village. For example, if he found out that the village leaders were involved in unethical or immoral activities that were against his values, it could make him hate Konoha. He might also feel betrayed if his so - called friends turned their backs on him for some reason.
In this fanfiction, Naruto might leave because of the political situation in Konoha. There could be factions that he doesn't agree with or that are causing unrest. Naruto, being someone who values peace and unity, might find it difficult to deal with the internal politics and decides to leave in order to find a more harmonious place or to figure out a solution away from the chaos.
Perhaps it's an old acquaintance from Percy's past who he reconnects with after leaving camp. They fall in love and she becomes the mother of his daughter.
Perhaps he feels pressured in Konoha. Having a kid adds more responsibility, and he might think that leaving Konoha will give him the space he needs to grow as a father and a ninja without the constant eyes of the village on him. He could be seeking a place where he can train his kid in a different way, away from the traditional Konoha methods.
Well, perhaps Naruto could be framed for a crime in Konoha and then be forced to leave as an assassin. He might start working for an enemy village at first, but then gradually realizes that there's a deeper conspiracy behind his framing. So he decides to clear his name and also protect Konoha in the end.
Well, it could be that Naruto, with his godlike powers, leaves Konoha because he feels misunderstood. Maybe the villagers still fear his power despite all he has done for them.