One possible motive could be that Minato is under some sort of dark magic or mind control. In fanfiction, authors often play with the idea of characters being forced to act against their true nature. So, if Minato is being controlled, he might end up doing something as extreme as killing Naruto.
The implications would be huge. It would completely shatter the positive family image that the Naruto series usually portrays. Fans who love the relationship between Naruto and his parents would be very shocked and possibly offended.
It's a rather unique concept. In the original Naruto storyline, Minato is Naruto's father and a highly respected figure. So a fanfiction where Naruto kills Minato goes against the typical father - son relationship and the positive image of Minato. It might explore darker themes such as extreme emotional turmoil in Naruto or some kind of mind - control that forces him to do such a heinous act. However, it's not a common theme as most fanfictions tend to follow the general positive and growth - oriented narrative of the original series.
A possible motive could be that Naruto discovers some dark secret about Minato that makes him feel betrayed. For example, if he finds out that Minato was involved in some unethical experiments that led to the suffering of many people. Or perhaps Naruto is tricked by a powerful villain who manipulates his emotions and convinces him that Minato is the root of all his problems.
Well, in the original Naruto storyline, Minato is Naruto's father and he sacrificed himself to save the village and Naruto. So, in the fanfiction where Minato kills Naruto, it completely goes against this fundamental relationship. It changes the very essence of Minato's character from a self - sacrificing hero to a killer.
There might be a very dark and twisted fanfiction like that. It could be that Naruto is tricked into believing that Minato and Kushina are some kind of threat to the village or to his friends. So in a moment of confusion and misinformation, he takes drastic action. But this goes completely against Naruto's character in the original series.
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.
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 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.