Well, in such a fanfic, Naruto might gain some extremely powerful and unique abilities. Kaguya being his trainer could mean he'd learn ancient and forbidden jutsus. Maybe he'd also gain a deeper understanding of the chakra and the origin of the ninja world.
One possibility is that Naruto, being neglected by the village, would be full of pent - up anger and a thirst for recognition. Kaguya's training could focus on unlocking hidden powers within him. For example, she might teach him ancient jutsu that no one in the village knows. This could lead to Naruto developing a unique fighting style that combines his Uzumaki heritage with Kaguya's otherworldly knowledge.
Well, Naruto would likely gain some really unique and powerful abilities. Since Kaguya is extremely powerful, she could teach Naruto some ancient and secret techniques that are not known in the normal ninja world. Maybe he'd learn more about the origin of chakra and how to manipulate it in ways he never thought possible.
One possible plot could be that Naruto, under Kaguya's training, gains godlike powers at an accelerated rate. He might learn ancient and powerful jutsu that were once exclusive to Kaguya. This could lead to him facing new and extremely powerful enemies, perhaps even threats from other dimensions. With Kaguya's guidance, Naruto could also gain a deeper understanding of the chakra and its origins, and use this knowledge to reshape the ninja world for the better.
One potential antagonist could be those who fear Naruto's new - found godlike powers. There might be ninja from other villages who see Naruto as a threat to the balance of power. Another could be remnants of Kaguya's old enemies who resurface, seeing Naruto as a continuation of Kaguya's influence and wanting to stop him. Also, there could be dark forces within the ninja world that want to corrupt Naruto's new powers for their own evil purposes.
His understanding of the ninja world would expand. He might become more cautious and calculating in battles, as Kaguya's training could expose him to different strategies. Also, he could develop a more solemn side to his personality, considering Kaguya's own complex nature and the heavy knowledge she might share.
Kakashi can be a popular character too. His wisdom and combat skills can be useful in fanfic scenarios involving Kaguya. He could be a mentor figure to Naruto, guiding him in dealing with Kaguya - related threats or helping to decipher ancient ninja knowledge related to Kaguya.
In this fanfiction scenario, Naruto and Hinata would likely gain extraordinary powers. Kaguya has immense and unique abilities. Naruto might learn new forms of chakra control or even some ancient jutsu that Kaguya knew. Hinata could improve her Byakugan skills to a whole new level, perhaps even being able to see things on a different plane of existence. They might also gain a deeper understanding of the origin of chakra and the history of the ninja world from Kaguya directly.
In this fanfiction, Naruto would likely have a huge power boost. Since Kaguya has such vast knowledge and power, she could train Naruto to better control his already massive chakra reserves. Maybe she'd show him how to use chakra in ways he never thought of before. For example, she could teach him to infuse his rasengan with some of her own unique chakra elements, creating a new and super - powerful attack. Also, Naruto's understanding of the origin of ninjas and the world itself could deepen through Kaguya's teachings.