Well, if Naruto and Kushina left Konoha in fanfiction, it would change a great deal. Naruto would not have the same relationships with his friends like Sasuke and Sakura. He might develop different fighting styles as he wouldn't be trained by the same people in Konoha. Kushina, being a strong kunoichi, could train Naruto in her own unique ways. And their departure could also cause a stir in Konoha, with some people trying to find them and others realizing the mistakes they made in how they treated Naruto and his family in the past.
The plot could develop in various ways. For one, Naruto's power growth might accelerate as he and Kushina explore different regions and learn new techniques. They could also encounter old enemies of the Uzumaki clan, which would add a new layer of conflict. Maybe they'd find a lost Uzumaki village and help revive it.
One possible reason could be for his safety. Konoha is a well - protected village with many skilled ninja. She might believe that leaving him there would ensure he has the best chance at survival and growth as a ninja. Also, she may have some mission or obligation that forces her to leave him behind, perhaps something related to the greater good of the village or the ninja world in general.
The relationship between Konoha and the Uzumaki clan might be further strained. With Kushina leaving, Konoha would lose an important link to the Uzumaki. And other Uzumaki members might see this as a sign that Konoha is not a safe place for them.
He might be given special missions. For example, he could be tasked with training the new generation of ninjas. His experience and skills would be invaluable in this role.
In this kind of fanfiction, Naruto might experience a very emotional moment. He could finally get to know his mother better. Kushina would probably embrace Naruto tightly, telling him how much she has missed him all these years. Maybe she would share some of the Uzumaki clan's secrets with him that no one else knew.
One possibility in such a fanfiction scenario is that Kushina's own chakra, which was already special as an Uzumaki, could have an impact on Naruto while he's sealed within her. It could either strengthen him in some ways or cause unforeseen consequences. Also, characters like Minato would have a different relationship with the situation. Instead of seeing Naruto as his son on the outside, he might be more focused on the presence of Naruto within Kushina, which could change how he approaches protecting his family and the village.
In this fanfiction, Naruto might finally get the motherly love he always craved. Kushina could share her unfulfilled dreams for Naruto, like seeing him grow up without the pain of being an orphan. They would probably have an emotional conversation about Naruto's father too.
In such a fanfiction, it would be a very dark and alternative take. Naruto, being godlike, might have been pushed to the edge by something in Konoha, like betrayal or unjust treatment. Maybe he discovers some dark secrets about the village leaders that make him snap.
In a fanfiction, if Minato were to slap Kushina, it would be a huge shock to the Naruto - Kushina relationship. It could potentially cause a rift between Naruto and his father Minato in Naruto's mind. Also, Kushina would likely be deeply hurt and angry, which might lead to her distancing herself from Minato. This event could also change the way Naruto views his family's past and his parents' relationship.
One possible scenario in the 'Kushina Alive Teaches Naruto' fanfiction is that Kushina teaches Naruto how to better control the Nine - Tails' chakra. Since she was the original Jinchuriki of the Nine - Tails, she has a deep understanding of its power. She could guide Naruto in using the chakra more efficiently and without the negative side effects that he often faced. This would be a huge boost to Naruto's abilities and his overall development as a ninja.