Well, it could be really interesting. Maybe Gaster would teach Naruto some unique and powerful magic or abilities that are different from the normal ninja skills in the Naruto world. Naruto might grow up with a very different perspective on power and friendship.
Naruto's personality could be a blend of his original upbeat and determined self with Gaster's more enigmatic and calculating nature. He might be more secretive about his plans and abilities. For example, instead of shouting about his goals all the time like in the original Naruto, he would keep them to himself and work quietly towards them. He could also be more curious about the unknown and scientific aspects of things, which is something he would pick up from Gaster.
I'm not sure if there is such a specific fanfiction. There are countless fanfictions out there, and while crossovers can be really creative, this one seems rather unusual. It would be quite interesting to see how the Naruto universe and the concept related to Gaster could be combined, but it might be a rare find.
In such a fanfiction, Naruto could be in a situation where he enters a dimension similar to the Undertale world where Gaster exists. He could learn about Gaster's experiments and try to use that knowledge to better his own village. For example, if Gaster had some unique way of manipulating energy, Naruto could attempt to adapt it to chakra manipulation. The story could also explore how Naruto's values from his own world interact with the often strange and complex ideas associated with Gaster.
Well, Naruto might become more mysterious. Since Gaster is an unknown and rather enigmatic entity, Naruto could start having strange thoughts or powers that he doesn't fully understand at first. He might seem more withdrawn as he tries to figure out what's going on inside him, and his normal cheerful and straightforward personality could be clouded by the presence of Gaster.
He could gain the power of manipulation. Gaster might give Naruto the ability to manipulate space or time on a small scale. Maybe he could create small portals or slow down time for a short period during combat.
One common theme could be the exploration of their relationship in the context of the outertale universe. It might involve how they interact with the unique settings and other characters in that AU.
There could be some. You might want to check out fanfiction websites like Archive of Our Own. However, the availability and quality can vary. Sometimes it takes some searching through different tags and filters to find the really good ones.
One possible scenario in this fanfiction could be that Naruto grows up with a more balanced upbringing. Tsunade might teach him medical ninjutsu and how to control his emotions better. Jiraiya, on the other hand, would surely train him in powerful combat techniques and in the ways of the sage. Naruto would probably be more mature and well - rounded at an earlier age, with a combination of Tsunade's tough love and Jiraiya's adventurous spirit guiding him.
One possible scenario in this fanfiction could be that Naruto grows up with the Namikaze Clan's unique jutsu training methods. He might inherit some of Minato's (a Namikaze) speed and intelligence in combat at a much earlier age. This could lead to him being a more confident and skilled ninja from the start, as the Namikaze Clan has a reputation for their strong abilities.