Well, it could be that Naruto, under Minato's training, quickly masters powerful jutsus. He might learn advanced space - time ninjutsu faster than in the original story. With Minato's guidance, Naruto could also develop unique combat strategies, making him seem godlike in battles against tough opponents like the Akatsuki members.
The Akatsuki could still be the main antagonists. Even with Naruto being trained by Minato and becoming godlike, the Akatsuki's goals of collecting the tailed beasts would put them in direct conflict with Naruto. Their powerful members, like Pain or Itachi, would pose a great challenge to Naruto, despite his new skills.
In this fanfiction scenario, Naruto's training with Minato and Kushina would be intense. Minato, being a prodigy, would push Naruto to his limits in terms of combat skills and quick decision - making. Kushina would focus on Naruto's inner strength and the connection to his Uzumaki roots. This could result in Naruto having a more well - rounded set of skills from the start. He might not struggle as much with basic jutsu control and could potentially even master some advanced Uzumaki - specific techniques earlier. For example, he could have a better grasp on sealing techniques which are a big part of the Uzumaki clan's abilities. This would give him an edge in battles against opponents like Orochimaru or the Akatsuki members who often use or are affected by sealing jutsu.
Naruto's personality would be a blend of his own natural exuberance and the more composed natures of Minato and Kushina. He might be less impulsive as Minato's strategic thinking would influence him. Also, Kushina's strong - willed nature would make him more determined and less likely to be swayed by others' negative opinions.
Their relationship would become closer. Naruto would finally have the father - son bond he always desired. Minato would be able to pass on his wisdom and skills directly to Naruto, which would make Naruto feel more connected to his family.
One interesting plot development could be Naruto quickly mastering a new jutsu that Minato was famous for, like the Flying Thunder God Technique. This would give Naruto an edge in battles and also create a connection between him and his father on a more practical level.
In fanfiction, Naruto and Minato often represent a connection between generations. Minato was a great ninja in his time, and Naruto is following in his footsteps. Fanfictions can explore how Naruto lives up to his father's reputation.
I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide variety of themes and scenarios, some of which might be rather strange or controversial. However, Naruto fanfiction often explores different relationships and alternate storylines, but this particular one seems a bit unorthodox and I haven't personally come across it.
One popular theme is Minato getting to know Naruto in the afterlife. Since Minato died when Naruto was a baby, fanfictions often explore their relationship in a spiritual realm.
One of the main similarities is their self - sacrifice. Minato sacrificed himself to save the village and his son. Naruto has also shown that he is willing to put himself in harm's way for the greater good. He's always the first to jump into a dangerous situation if it means protecting those he loves. Another aspect is their leadership qualities. Minato was a great leader as Hokage, and Naruto has developed into a great leader as well, inspiring those around him.