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.
In this fanfiction, Naruto might finally get the fatherly guidance he always craved from the Fourth Hokage. Maybe the Fourth Hokage would teach Naruto some long - lost jutsu that only he knew. It could also be a chance for Naruto to understand his father's sacrifices on a deeper level.
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.
Naruto himself is often a popular character in these fanfictions. Since he is the main character of the original series and has a connection to the Fourth Hokage as his father, his presence adds depth to the story. Kakashi is also common. His relationship with the Fourth Hokage was that of a student - teacher, and it can be further explored in fanfictions.
One option is a new, original villain created just for this fanfiction. This villain could be someone who wants to rewrite history and prevent the Fourth Hokage from ever becoming Hokage. So they target Naruto as he is the son of the Fourth Hokage. They might have some sort of time - traveling or reality - warping powers.
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.
It could strengthen Naruto's relationship with Iruka. Since Naruto is the son of the fourth hokage, Iruka might see Naruto as an even more important part of the village and be even more protective. For example, he could be more involved in Naruto's advanced training.
His relationship with Sasuke would be very different. Sasuke might see Naruto in a new light, perhaps with more respect as he's achieved the highest position in the village. Naruto would also have a different dynamic with Kakashi. He'd be more of an equal now, and Kakashi would have to adjust to Naruto being in a leadership role.