If Naruto could turn invisible in the fanfic, his relationships would experience a great deal of upheaval. His peers in the village might start to view him as more of an enigma. Some might be in awe of his power, while others could be a bit intimidated. His relationship with the older generation, such as the village elders, could become more complex. They might see him as a greater asset but also a potential threat if his power gets out of control. His relationship with Hinata could also take an interesting turn. Hinata might be shy as always but also more curious about Naruto's new ability, and Naruto could use his invisibility to playfully interact with her in a new way.
His relationship with Sasuke might be affected. Naruto might approach Sasuke differently, with new strategies learned from Zetsu. He could be more secretive about his intentions towards Sasuke, which might either make Sasuke more curious or more distrustful. With his friends in Konoha, they might be a bit wary of Naruto at first because of his new training source. But once they see that Naruto is still on their side, they would support him.
His relationship with the Hidden Leaf would be strained. He might see the village as a source of weakness rather than a place to protect. And his friends in the village, like Sasuke and Sakura, would be more like strangers to him. He would not have the same bonds that were formed through growing up in the Leaf.
Naruto's relationship with his sensei, like Kakashi, would change. Kakashi would be both proud and cautious. Proud because Naruto is getting stronger, but cautious as the Shinigami - related training is uncharted territory. His bond with the other villagers would also shift. Some might see him as a protector with new, otherworldly powers, while others might be suspicious of the Shinigami's influence on him. Overall, his relationships would be in a state of flux as he adapts to his new self after the training.
With his friends in Konoha, like Sakura and Kakashi, there would be a major shift. Sakura might initially be conflicted between her loyalty to the village and her friendship with Naruto. Kakashi, being a more experienced shinobi, might understand Naruto's reasons better but still be bound by his duties to the village. Overall, Naruto's departure as a nukenin would create a lot of emotional turmoil among his relationships.
If Naruto is in a female - ruled fanfic, his relationships would shift. With female characters in power, he would have to interact with them differently. For instance, his friendship with Sakura could change as she might have a more authoritative role. And his relationships with female antagonists could also be different as they might be more in line with the female - led power structure in the fanfic.
His relationship with Sasuke would be deeply affected. Without being able to verbally challenge Sasuke or express his feelings directly, Naruto would have to rely on his actions even more. Sasuke might initially misinterpret Naruto's silence as weakness or indifference.
If Naruto becomes cold, his relationship with Hinata would be severely affected. Hinata, who loves him for his warm heart, might be at a loss. Naruto might start to push her away, not allowing her to get close. His friends from Team 7 might try to bring back the old Naruto, but his coldness would create a rift between them.
His relationship with his old friends like Sasuke would likely become more of an equal partnership. Since they are both highly skilled, they could collaborate on important missions. With Sakura, he might be seen as more of a respected colleague, and she could look up to him in a different way as a Jounin.
His relationship with Sasuke would be greatly affected. Sasuke might see Naruto's move as a betrayal to Konoha and their bond. On the other hand, Naruto's relationship with Konan would become very strong as they are now comrades in Amegakure. He might also form new friendships with the shinobi of Amegakure who would look up to him for his power and his new ideas for the village.
His relationships with his old friends in the Hidden Leaf would become extremely strained. They would see him as a traitor. For example, Sakura might be heartbroken and angry at him. Kakashi would be disappointed. However, within Akatsuki, he could form new relationships. He might become friends with some of the members who have a similar background of being misunderstood. For instance, he could bond with Hidan over their strong beliefs, though their beliefs are very different in nature.