Well, if Naruto goes Super Saiyan 3 in a fanfiction, he would likely gain an insane amount of power. His physical abilities would skyrocket. He could move at incredible speeds, and his strength might be enough to shatter mountains. His chakra would probably combine with the Saiyan power in some unique way, creating new and powerful jutsu.
Well, in this fanfiction concept, Naruto going Super Saiyan 4 could also lead to some interesting character dynamics. His friends might be initially shocked and a bit scared of this new form. Naruto would then have to prove that he's still the same person despite this radical transformation. And as for the enemies, they would surely be caught off - guard and have to quickly come up with new strategies to deal with this super - powered Naruto.
Vegeta would also be an important character. His rivalry with Goku would intensify even more. Seeing Goku achieve the Super Saiyan form early would fuel his jealousy and his desire to become stronger. He might even try to find new ways to train or gain power to catch up with Goku.
In a Naruto goes Super Saiyan Rose fanfiction, the story could start with Naruto feeling a strange energy within him. This energy could be related to the Super Saiyan Rose form. As he tries to understand it, he discovers that it gives him enhanced speed, strength, and a new set of abilities. He then has to learn to control it while also dealing with the reactions of his friends and the ninja world around him. For example, Sasuke might be initially suspicious of this new power Naruto has acquired.
One common plot is Naruto using his Super Saiyan power to defeat enemies that were previously too strong for him. For instance, Orochimaru might seem like a pushover once Naruto can go Super Saiyan. Another plot is about Naruto trying to hide his new power at first, fearing how others will react. Then, when the situation gets dire, he reveals it and saves the day. It's like a classic superhero - type plot.
Perhaps in the fanfiction, Goku goes Super Saiyan early due to some special training or a unique emotional trigger. It could be that he witnesses the death of a different character, like Krillin, much earlier than in the original story. This extreme emotional distress causes him to break through his limits and transform. As a result, the Saiyan saga would take a different turn. Frieza might become aware of Goku's new power much sooner and plan different strategies to deal with him.
In these fanfictions, Naruto's transformation into a Super Saiyan can change the whole dynamic of the Naruto world. First, his appearance would be quite shocking to those around him. His eyes might take on the Saiyan - like look, and his overall aura would be different. Story - wise, he could start training with his new form, learning how to better control the Super Saiyan power. This might involve him going on a journey similar to Goku's in Dragon Ball, seeking out stronger opponents to test himself. He could also teach his friends some of the Saiyan fighting techniques, like the Kamehameha - like attacks that could be modified to fit the Naruto universe.
Naruto's personality could change in several ways in this fanfiction. Firstly, he might become more impatient. The Super Saiyan Rose power is extremely powerful, and he may want to use it right away to solve problems. Also, he could become more protective. Since he has this new and powerful form, he would feel a greater responsibility to protect his friends and the village. However, he would still retain his core values of friendship and loyalty. He might just be a bit more headstrong when it comes to defending those he cares about.