Yes, in many fanfictions, Naruto is often depicted as a huge fan of the Fourth Hokage. After all, the Fourth Hokage is his father. Naruto grows up idolizing the Fourth Hokage's heroic deeds and wishes to be like him.
One interesting Naruto Fourth Hokage fanfiction could be one where he survives the fight against the Nine - Tails. In this kind of fanfiction, he gets to see Naruto grow up and train him personally. It would be a heartwarming story of a father - son relationship that was cut short in the original series.
Kushina is an obvious choice. In fanfictions, they can have more adventures together, or their relationship can be explored in more depth. Their love story is already quite popular in the Naruto series, and in fanfictions, it can be expanded upon, like showing more of their courtship days.
In this fanfiction, the plot might involve Naruto being tricked by an evil force into believing that the Fourth Hokage has some dark secrets. So he decides to fight him. It could also be a scenario where Naruto enters a different dimension where the Fourth Hokage is being controlled by an outside power, and Naruto has to free him by fighting. And during the fight, Naruto discovers new things about his father and his own abilities.
There's a fanfic where Naruto starts dating a girl from another village. The Fourth Hokage, in a spiritual form, is always watching over him. The fanfic is called 'Naruto's New Love with Yondaime's Protection'. It has a great plot that weaves in the Fourth Hokage's protective nature and Naruto's journey in love.
In a fanfiction battle, perhaps the fight begins with Naruto trying to talk to Goku first, to understand his intentions. Goku, not really understanding Naruto's world, just wants to test his strength. So he attacks without warning. Naruto quickly enters his Nine - Tails Chakra Mode to defend himself. The fight then escalates as Goku powers up further. The two exchange blows, with Goku's punches creating shockwaves and Naruto's attacks leaving marks on the ground. As the fight goes on, they both start to respect each other's power and skills, and it becomes more of a battle of wits as well as strength.
One possible plot twist could be that Naruto has to face an unexpected enemy from the past of the Fourth Hokage. Maybe a hidden group that was against Minato's policies and now they see Naruto as a threat. Another could be that Naruto discovers a new jutsu that Minato was developing but never completed, which changes the power dynamics in the village.
It's really hard to say. The Fourth Hokage was incredibly powerful, with his Flying Thunder God Technique and all. But Naruto has the Nine - Tails' power and has grown a lot. If it was a fair fight, it might depend on who has the better strategy. The Fourth is more experienced in combat, but Naruto has that raw power. So it could go either way.
One interesting plot twist could be that the fourth hokage didn't actually die when he sealed the nine - tails in Naruto. Instead, he was trapped in a sort of limbo within Naruto's mind and could communicate with him at crucial moments, guiding him in ways that no one else could understand.
Another scenario could be during battles. Naruto might be inspired by the Fourth Hokage's battle strategies. He may think to himself, 'What would the Fourth Hokage do in this situation?' and then try to act accordingly. In some fanfictions, Naruto even visualizes the Fourth Hokage beside him, giving him advice during tough battles.