Well, one way to explore their relationship is to create a scenario where Minato is still alive in a parallel world and is raising toddler Naruto. We can see how Minato would teach Naruto about his heritage, the Uzumaki clan, and how to control his chakra. We could also explore their relationship through the eyes of the villagers. Since Minato was a hero, how do they react to seeing his son, toddler Naruto, and how does Minato protect Naruto from any negative attention? Also, we can have Minato tell Naruto stories about his own adventures as a ninja, which would strengthen their bond.
Their relationship can be explored through flashbacks. For example, Minato could have left behind memories or messages for toddler Naruto that are revealed as Naruto grows. Another way is to have Minato's spirit interact with toddler Naruto, like guiding him in his dreams. They could also explore their relationship by showing how Minato's traits are already starting to show in toddler Naruto, such as his determination.
One way to explore their relationship is to have Minato's friends, like Jiraiya, tell baby Naruto stories about his father. As Naruto listens, he starts to form an image of Minato in his mind. This could lead to Naruto trying to imitate Minato in different ways. Also, the fanfic could show how Minato's sacrifice for the village is slowly understood by baby Naruto as he grows up, deepening the emotional connection between them.
Well, their brotherly relationship could also be shown in how they deal with the village's problems together. Naruto's strong will and Minato's wisdom could combine. They might face off against a new, powerful enemy that threatens Konoha. Naruto would be all about charging in, while Minato would plan a more strategic approach, and they'd have to learn to work with each other's styles.
Their relationships could be explored through flashbacks. For instance, we could see how Minato interacted with Tsunade when he was alive. Maybe he sought her advice on certain matters, or they had some joint missions. This would give us insights into their respect for each other.
The relationship can be developed by having Minato appear in Naruto's memories more vividly. For example, Naruto could start having flashbacks of Minato's training sessions or family moments that he didn't know about before. This would make Naruto feel closer to his father.
Well, in some fanfictions, instead of the typical rivalry starting early, Naruto could be like an older brother figure to toddler Sasuke right from the start. He could teach Sasuke simple things like how to climb a tree or catch a fish, which would build a strong bond of trust and friendship between them.
The emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
It would be very close. Naruto has always longed to know his father, so when he travels back in time to meet Minato, they would quickly bond. They'd probably start training together right away.
Another aspect could be a time - travel fanfiction. Naruto gets sent back in time and meets a young Minato. They train together, and Naruto shares some of his future knowledge in a veiled way. This could lead to a different development of Minato's skills and a unique bond between them.
Fanfiction can explore their relationship by delving into their unfulfilled time together. For example, showing how Minato and Kushina would have raised Naruto. It could also explore their shared experiences, like how they all have a connection to the Nine - Tails. Maybe in fanfiction, they have family adventures where Naruto learns from his parents' skills and wisdom.
The relationship can start with mutual respect. For example, if female Naruto shows great courage in a battle, Minato, who values bravery, will start to notice her. Then they can have small interactions like sharing their thoughts on ninja philosophy which will bring them closer.