His character could become more complex. He might start to question his values as the power from the gate of death could be a corrupting influence.
Naruto's character might go through a major transformation. The experience of opening the gate of death could make him more cautious, as he realizes the great risks associated with such power. He could also become more empathetic towards those who have been corrupted by power in the past. His relationships with his friends like Sasuke and Sakura would change significantly. He might distance himself from them at first to protect them from the potential danger he now poses, but as he learns to control this new power, he could come back to them with a new - found wisdom and strength.
In fanfiction, if Naruto were to open the gate of death, he might gain an extreme power boost. This could potentially allow him to defeat enemies that were previously unbeatable. However, it would likely come at a great cost, perhaps his own life or a part of his soul.
His character could become more complex. As a homunculus, Naruto might struggle with the new moral implications of his existence. He was always a positive and determined ninja, but now he may have to face the darker side of his new form.
His character would become more confident. The power of the Flash would make him realize his potential even more. He'd be less likely to doubt himself in tough situations.
Naruto's character could become more complex. He'd have to deal with the moral implications of using Jiongu. Since Jiongu has some rather strange and powerful abilities, Naruto would need to ensure he doesn't misuse it. This would add depth to his character as he'd be constantly making decisions about when and how to use this new power.
Naruto's character could develop morally if he finds Mjolnir. Since Mjolnir is only liftable by those who are worthy, it would reinforce Naruto's sense of justice. He'd see himself as even more of a protector. He might start to view his ninja way in a new light, perhaps incorporating some of the values associated with Mjolnir. For example, he could become more strict about who is worthy of his protection, like Thor is with Mjolnir.
His character would become more cerebral. Instead of relying solely on his physical strength and ninjutsu, he'd use his intellect. He might become more patient as scientific research requires time.
If Naruto joins Iwa in fanfiction, his character development could be significant. He'd have to deal with being an outsider at first. This would test his resolve and make him grow in patience. He may also start to see the world from a different perspective, as Iwa's goals and values might be different from Konoha's. This could lead to him becoming more diplomatic and understanding of different ninja cultures.
If Naruto got Wolverine's power in fanfiction, his development would be really interesting. He'd have to learn to control the healing factor and not rely too much on it. His sense of self - worth might change as he realizes he has this new, extremely powerful ability. He might also become more of a loner in some ways, but still be part of the Konoha community. He could start to question his ninja way in the face of this new power, but ultimately find a way to incorporate it into his values.
Naruto's character could develop in terms of his leadership skills. As a Greek god, he'd likely have to lead or be part of the godly community. His natural leadership from being a ninja could be enhanced. He'd also have to learn to control his emotions better, as Greek gods often have tempers, and Naruto would need to find a balance between his old self and his new godly self.
Naruto's character could develop a greater sense of self - worth. Knowing his parents and their love for him directly would boost his confidence. He might become more mature, as he now has a connection to his family roots.