In a fanfic where Naruto is trained by Zetsu, the story could take an interesting turn. Zetsu has knowledge of the Akatsuki and their secrets. Naruto could learn about the organization from the inside out. This knowledge could help him in his quest to bring peace. He might also gain some physical enhancements. Zetsu is quite durable in his own way, and Naruto could inherit some of that sturdiness. This would make him an even more formidable opponent in battles against strong foes like Pain or Madara.
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.
Well, one possibility could be that Zetsu imparts some unique and dark jutsu to Naruto. Maybe Naruto starts to learn stealth and subterfuge skills from Zetsu, which are different from what he learned at the ninja academy. This could lead to Naruto having an edge in certain battles where traditional ninja skills aren't enough.
There could be internal antagonists within Naruto's own village. Some of the more conservative ninja in the village might not trust Naruto's new - found skills and powers obtained from Zetsu. They could try to undermine Naruto or even attempt to exile him, fearing that he has been corrupted by Zetsu's teachings.
It's likely that Naruto's chakra control and power would develop in new and unexpected ways. Black Zetsu might introduce him to strange chakra - based abilities that are not part of the normal ninja training. This could give Naruto an edge in combat, but it also might make him a target. Since Black Zetsu has his own motives, Naruto could be dragged into some dangerous and complicated situations that he may not be fully prepared for.
Well, one way could be in their physical abilities. Maybe in the fanfic, Naruto is shown to have some of Zetsu's stealthy movement skills, like being able to blend into the environment in a similar way. Another similarity could be in their connection to nature. Zetsu has a strong link to the plant - like aspects of nature, and Naruto's chakra might be shown to have a similar kind of connection in the fanfic.
He may develop a more calculating side. Black Zetsu is all about manipulation. Naruto could start thinking several steps ahead in a more devious way. For example, he might start using people's weaknesses against them, which is completely against his original nature. This change in personality would likely cause a rift between him and his friends who know him as the always - honest and straightforward Naruto.
The fanfic might explore new aspects of Naruto's character by comparing him to Zetsu. It could show different sides of Naruto that we haven't seen before, like his more devious or strategic side if he takes on some of Zetsu's traits. This exploration of character depth can be really engaging for readers.
Perhaps Naruto controls Zetsu to help him in battles. Zetsu could act as a distraction while Naruto sets up his more powerful jutsu. For example, Zetsu could split into multiple forms and confuse the opponents, giving Naruto an opening to attack.