One way could be that Vegeta uses the knowledge from Goku's memories to find new ways to train. He might adapt some of Goku's training techniques to improve his own power level.
It's possible that Vegeta would gain a new respect for Goku. By seeing Goku's memories, he gets to understand the full scope of Goku's life, his struggles, and how he overcame them. This could lead Vegeta to view Goku in a different light and perhaps even strengthen their rivalry in a more positive way.
In some fanfictions, it might be a really strange and unique concept where Goku is depicted as Vegeta's pet. But this is far from the original relationship in the official Dragon Ball story. It's just a creative and often humorous take that some fanfic writers explore.
Yes, in the vast world of fanfiction, there are all kinds of creative and out - of - the - box stories. Some fans like to explore unusual relationships and scenarios, so it's possible that there is such a fanfiction out there. However, it goes against the original canon of the Dragon Ball series in many ways.
Bulma could be involved. Since she has a relationship with Vegeta and is friends with Goku, she might try to mediate between them when Goku snaps at Vegeta. She could be the voice of reason, trying to calm both of them down and make them see each other's point of view.
His power would develop more in line with Saiyan traditions. He'd probably have a stronger base power due to the intense training environment on Planet Vegeta. There would be more emphasis on raw power and combat skills from a young age.
Well, he would probably be a very different person. He might be more in tune with the Saiyan warrior culture. He could be more aggressive and less empathetic compared to the Goku we know. He would grow up learning the ways of the Saiyans on Planet Vegeta, which means he'd be trained to fight and conquer from a very young age.
Many Goku and Vegeta fanfictions explore the concept of power. How they strive for more power, what they are willing to sacrifice for it, and how they handle having great power. This can lead to some interesting moral and ethical dilemmas within the stories.
There might be some out there, but such a concept goes against the more common friendly - rival dynamic between them in the original series. It's not a very popular or positive take in the mainstream fan community. However, in the vast world of fanfiction, you might find some that explore this in a creative, non - malicious way, perhaps as a what - if scenario in an alternate universe where their relationship took a different turn.
In some fanfictions, Vegeta could be under the influence of some external force. Maybe an evil entity has convinced Vegeta that kidnapping Goku is the key to achieving ultimate power. Vegeta, being power - hungry as he is, falls for it and kidnaps Goku. But later, he realizes his mistake and has to make amends, which can lead to an interesting story arc of redemption and friendship building between the two Saiyans.