Well, his relationship with Kakashi would shift. Instead of being just a student, Naruto would be more like a peer in terms of status, as he's now the Sandaime's apprentice. This could lead to new forms of cooperation between them. Naruto would also form new relationships with the Sandaime's old friends and allies in the village. He might get to know some of the retired ninja who were once close to the Sandaime and gain their support and wisdom, which would further shape his character and his relationships in the village.
Naruto's relationship with the villagers would improve more rapidly. The Sandaime could help Naruto show the villagers his growth and potential in a more organized way. Naruto would also have a different relationship with the other ninja. He'd be seen as someone with a closer connection to the Hokage, so they might be more respectful or more likely to follow his lead. And his relationship with his own inner self would change too. He'd have more confidence knowing he's being trained by such an important figure.
His relationship with Sasuke could become more competitive in a different way. Instead of just competing in ninjutsu, they would now compete in swordsmanship skills. This could add a new layer to their rivalry.
One possible scenario is that Naruto would gain a lot more knowledge about the village's history and politics. The Sandaime, being an experienced Hokage, could teach Naruto about the different clans in the village, their relationships, and how to manage them. Naruto might also learn more advanced jutsu more quickly, as the Sandaime could give him personal training. This could make Naruto a much stronger ninja at a younger age and perhaps change the course of some of the earlier story arcs.
His relationship with Sasuke would be really interesting. Since Sasuke has his own complex connection to the Uchiha clan and Madara, Naruto becoming a Madara fan could either bring them closer together in understanding the Uchiha's past or drive them further apart if Sasuke doesn't agree with Naruto's new - found admiration. Maybe Naruto would start to push Sasuke towards a more 'Madara - like' view of power and revenge.
His relationship with Sakura would change a lot. Sakura might be initially scared of Naruto's fox form, but as she realizes it's still him inside, she would start to see him in a new light. She could become more protective of him. With Kakashi, Kakashi would be more cautious and observant, trying to figure out how to help Naruto control his new form. And with his rivals, they might either be more envious or more afraid of him now that he has this unique transformation.
His relationship with Sasuke would be very different. Sasuke might see Naruto as even more of an enigma. He could be both jealous of Naruto's new power and also concerned about what Naruto has become. Naruto, on the other hand, would try to show Sasuke that he's still the same person deep down.
Naruto's relationship with the villagers could be affected. As he becomes a Minato fan, he would try to uphold Minato's legacy even more. The villagers, seeing Naruto's newfound admiration for Minato, might start to see Naruto in a new light, more like how they saw Minato. Also, his relationship with Hinata could be different. Naruto might start to model his family - centered values after Minato, which could bring him and Hinata even closer as they plan their own family.
His relationships would change drastically. With his friends in the Hidden Leaf, there would be a sense of distance. For example, Sasuke might be confused by Naruto's decision. But at the same time, Naruto might form new relationships with the animals and plants in his hermitage area.
His relationship with Sasuke would become very strained. Sasuke, who has always been somewhat distrustful, would see Naruto as a real threat now. He might even consider Naruto an enemy and try to stop him.
Naruto's relationship with the other Leaf Village ninjas would also shift. Some might be suspicious of his new - found admiration for Itachi, especially those who were directly or indirectly affected by Itachi's actions. However, others who understand Itachi's true motives, like Kakashi, might support Naruto. This could create a complex web of relationships within the village, with Naruto at the center trying to balance his new beliefs and his old friendships.
His relationship with Sasuke would be severely strained. Sasuke, who has his own goals and views on power, would see Naruto's demon transformation as either a threat or a sign of weakness. He might distance himself from Naruto or even see him as an enemy to be defeated.