Maybe Naruto is put in a situation where he feels that Hiruzen's way of handling things is the best option. For instance, if there is a major crisis in the village that requires a more traditional and experienced approach, Naruto might choose to act like Hiruzen. Also, he could be influenced by the stories he has heard about Hiruzen and wants to fill his shoes in a way.
It could be that Naruto is trying to bridge the gap between the old and new ways of the village. Hiruzen represented the old - school leadership, and Naruto, as the new - age hero, by acting like Hiruzen, can combine the best of both worlds. He can bring in his own ideals while also using Hiruzen's methods to ensure a smooth transition and better governance in the village.
Well, Naruto might start showing more wisdom like Hiruzen. He could become more diplomatic in handling situations. For example, when dealing with disputes among the villagers, instead of his usual impulsive reactions, he would think more like Hiruzen, considering the long - term effects on the village as a whole.
One reason could be for strategic purposes. Itachi was a highly strategic ninja. So, in fanfiction, Naruto might act like him to outwit a particularly difficult enemy. For instance, when facing an enemy who is a master of deception, Naruto could adopt Itachi's calm and calculating demeanor to see through the enemy's tricks.
Maybe Naruto admires Kakashi's skills. Kakashi is a highly respected ninja, and Naruto might want to be as good as him. So he starts acting like him to learn from him.
Maybe in the fanfiction, there were some political motives. Hiruzen might have been pressured by other hidden villages. They could have threatened Konoha if Naruto, as the Nine - Tails' jinchuriki, remained in the village. So, to avoid a potential war or conflict, Hiruzen made the difficult decision to banish Naruto. Also, it could be due to a misunderstanding. Naruto might have been in the wrong place at the wrong time during an incident, and Hiruzen misjudged the situation.
One reason could be the way Hiruzen treated Naruto as a child. In the original, Naruto was kind of an outcast, and if in fanfiction, Naruto blames Hiruzen for not doing more to prevent that, he could hate him. Another reason might be related to Hiruzen's decisions regarding the Uchiha clan. If Naruto in fanfiction feels that Hiruzen's actions led to bad consequences for the village and his friends, he might develop hatred.
In this fanfiction scenario, it could be set in an alternate universe where Hiruzen didn't die. Naruto, having grown powerful, might be in a conflict of ideologies with Hiruzen. Naruto, with his more modern and inclusive views, could clash with Hiruzen's more traditional ways. This leads to a fight where Naruto tries to prove his point through combat, using his new and improved jutsu that he developed through his own unique training methods.
Maybe for a new plot twist. It makes the story different from the original Naruto storyline.
Perhaps in this fanfiction, Naruto has some sort of special power or transformation that he unlocks just in time to face Orochimaru. It could be related to his Nine - Tails chakra in a new and unique way. He uses this power to break through Orochimaru's defenses and save Hiruzen. Naruto might also use his knowledge of Orochimaru's weaknesses that he has learned either from Jiraiya or from his own past encounters with Orochimaru's minions.
In a 'naruto adopted by hiruzen fanfiction', the plot could involve Naruto having a closer bond with the older generation of shinobi. Hiruzen would surely introduce Naruto to his trusted comrades. Naruto could learn unique jutsu from them. Also, his growth as a ninja would be different. Instead of seeking attention through pranks, he might focus more on living up to Hiruzen's name. His relationship with Sasuke could also be affected. Maybe they would become rivals in a more positive way, with Naruto having a stronger sense of family and purpose because of Hiruzen's influence.
One interesting story could be about Hiruzen mentoring Naruto in some unique jutsu that was long forgotten. This would show a different side of their relationship, focusing on the transfer of knowledge from the old to the new generation.