What upgrades or abilities would you add to the Byakugan to make it equal to or on par with the Mangekyo Sharingan?
Before we continue, I wanted to remind you that some of the ideas here I got from the internet, while some are my own. Please comment if you have any ideas or if you disagree with me; I would really appreciate feedback.
First, if you want to make the Byakugan as strong as the Sharingan, it needs the same evolution line:
Sharingan
Mangekyo Sharingan
Eternal Mangekyo Sharingan (not necessarily)
Rinnegan
RinneSharingan
The Mangekyo Sharingan is too OP, so the Byakugan needs to be OP in the same way. The Mangekyo Sharingan has OP abilities, with Susanoo being the most OP, so the Byakugan would need something to match it. Kishimoto never gave other kekkei genkai as much importance as Wood Release and the Sharingan, but I'm not here to talk about Kishimoto's mistakes. I'm here to brainstorm ideas.
Most abilities don't have names yet, but I will come up with suitable Japanese names. We know the Sharingan can cast Genjutsu. The Byakugan won't; instead, it will have unique abilities.
So, let's start with Kaguya, the mother of Hagoromo and Hamura Otsutsuki.
She had two dōjutsu:
RinneSharingan
Byakugan
All her eyes are classified as Kekkei Mōra.
The RinneSharingan split into two eyes:
Rinnegan
Sharingan
Then, the Sharingan evolved into the Mangekyo Sharingan.
There are different types of Mangekyo Sharingan for different users.
The Rinnegan was a natural eye for Hagoromo Otsutsuki, and his Rinnegan is considered kekkei mōra. He was the first human (half) to activate it and the first user of the Sharingan (not confirmed but could be).
He had a brother, Hamura Otsutsuki, who had the Byakugan. His Byakugan was a kekkei genkai, like the Sharingan (not confirmed) for Hagoromo. But Hagoromo had the Rinnegan, which was kekkei mōra, so we will give Hamura a kekkei mōra eye. Hamura acquired another eye, the Tenseigan.
Now, the Tenseigan is not like the Rinnegan (Tenseigan is kekkei genkai), but we will make it kekkei mōra to put both their eyes at the same evolution point.
To make matters simpler, we will give Kaguya a new kind of eye instead of the Byakugan: the TenseiByakugan, like the RinneSharingan.
That TenseiByakugan will split into the Byakugan and the Tenseigan, like the RinneSharingan split into the Sharingan and the Rinnegan.
Just as all the Otsutsuki clan members we've seen have two Byakugan and one or two Rinnegan, Kaguya will have one Rinnegan, which becomes the RinneSharingan, and two Tenseigan, which become the TenseiByakugan after eating the chakra fruit. The chakra fruit contains the genetic data of that planet and its living things, so all her abilities evolve:
Shikotsumyaku -> All Killing Ash Bones
Rinnegan -> RinneSharingan
Tenseigan -> TenseiByakugan
Tenseigan -> TenseiByakugan
The Byakugan won't be an Otsutsuki eye. Every Otsutsuki will have the Tenseigan instead.
(I don't know much about genetics and how they are passed down, but I will assume some things to make the answer simpler.)
Kaguya had five genetic traits after eating the chakra fruit (or before):
Rinnegan (became RinneSharingan)
Tenseigan (becomes TenseiByakugan)
Tenseigan (becomes TenseiByakugan)
Shinju Body (strong life force body) and Ash Bones
We don't know what happened to the Ash Bones and how the Kaguya Clan acquired them, but that's not the scope of this answer. Out of the four genetic traits, Hagoromo acquired the Rinnegan and Shinju Body, while Hamura acquired both Tenseigan. After splitting the genetic traits for the next generation, we get:
Because in the original, Indra had the Sharingan, Ashura had the Shinju Body.
The first child of Hamura with taijutsu-based Byakugan (more eye-related abilities)
The second child of Hamura with chakra-based Byakugan (more chakra-related abilities)
The third child of Hamura with both chakra and taijutsu-based Byakugan
The third child will start the Moon Otsutsuki, while the first two children will start the Hyuga clan.However, in this story, like Hagoromo, Hamura will also have two children. In the Naruto series, Sasuke and Naruto are said to be reincarnations of Indra and Ashura, having been reincarnated in several lives, such as Madara and Hashirama. Similarly, Hamura has two daughters named Sumire and Suiten. Sumire is the firstborn, a prodigy like Indra, while Suiten, like Ashura, struggles with everything.
Sumire and Suiten have been reincarnated alongside the brothers in several lives. Sumire's previous reincarnation is Mito Uzumaki, the second jinchuriki of the Ten-Tails and Hashirama Senju's wife. Suiten's previous reincarnation is Sayaka Hyuga, the firstborn daughter of the Hyuga clan leader and Madara Uchiha’s lover (during the Warring States Period). You might think Sumire and Indra are a pair, and Suiten and Ashura are a pair, but you're wrong. Indra and Suiten are a pair, and Ashura and Sumire are a pair. I won't spoil the sisters' next reincarnations; you'll find out in the story.
The Uchiha clan is descended from Indra, and the Senju-Uzumaki from Ashura, but here, the Uzumaki are not descended from Ashura. Instead, the Uzumaki clan is descended from Sumire, and the Hyuga clan from Suiten.
The two sisters will be like Ashura and Indra: not fighting, but having a deeper relationship with the brothers, especially Suiten and Indra. They will reincarnate (to maintain peace, as Hamura instructs them to help the brothers). Hagoromo, as the Ten-Tails' jinchuriki, had his powers split, giving Ashura the body of the Sage and Indra the eye of the Sage and the curse mark. Something similar will happen with Hamura's children.
Kaguya had two TenseiByakugan (now), while Hamura will have two Tenseigan, and Hagoromo had two Rinnegan and the body of the Shinju (strong life force body). Since Hamura had two Tenseigan (and not Shinju bodies), his two daughters will have different powers, like Indra had the Sharingan, Ashura had the Shinju Body. Suiten will have both Byakugan, while Sumire will have all the abilities of the Uzumaki clan, such as a strong life force, high chakra reserves, and mastery of fuinjutsu (sealing techniques).
Like the Rinnegan, achieving the Tenseigan requires two halves of Hamura's chakra. Like in the original, to awaken the Rinnegan, you need both Senju and Uchiha cells. But in this story, to awaken the Tenseigan, you need both Uzumaki and Hyuga cells.
In this scenario, there are four great dojutsu, not three: Sharingan, Rinnegan, Byakugan, and Tenseigan. The Tenseigan is a legendary eye, like the Rinnegan, and people know about Hamura as well as Hagoromo. Hamura will be called the Byakugan god (a Sage of Six Paths is unique, and I couldn't find anything like that).
Now for the evolution line of the Byakugan:
Byakugan extra abilities, like the Sharingan's Genjutsu abilities:
Uhijini/Tsunugui (or anything based on Japanese gods like Izanagi or Kotoamatsukami): This ability will block/open all 361 Tenketsu points in the Byakugan's field of vision at the cost of an eye, killing everyone except the user.
Suhijini/Ikagui (or anything based on Japanese gods like Izanami or Kotoamatsukami): This ability will block the Byakugan's field of vision (opponents), preventing them from using Uhijini/Tsunugui or any other Byakugan abilities until you have a change of heart. It can also work on non-Byakugan users, blocking their senses (eyes included), even though Senjutsu and Six Paths Senjutsu senses won't work.
These abilities don't rewrite destiny because the Byakugan can't cast Genjutsu, so I made it OP differently.
Then the Bankakyo Byakugan, the Mangekyo Sharingan version of the Byakugan. There will be normal Byakugan, then Bankakyo Byakugan. Acquiring it is simple: you need a strong sense of positive emotion, like joy, gratitude, hope, serenity, interest, pride, amusement, inspiration, and especially love. The Uchiha have a curse of hatred, while the Hyuga clan has a curse of love.
I will also change some of the Uchiha clan's Curse of Hatred. Uchiha need to feel negative emotions like fear, anger, sadness, disgust, jealousy, guilt, shame, loneliness, anxiety, disappointment, frustration, especially hatred. Positive emotions empower the Bankakyo Byakugan, while negative emotions empower the Mangekyo Sharingan.
Tobirama explained this: the Uchiha feel love stronger than any other clan. The more you love someone, the angrier you'll feel if they get hurt. Well, I will change that to: the more negative emotions an Uchiha has, especially hatred, the stronger they become. This also applies to the Hyuga clan, but oppositely: the more positive emotions a Hyuga has, especially love, the stronger they become. This will empower his/her Bankakyo Byakugan and give him/her a power boost to protect someone they love.
I changed the Uchiha clan's Curse of Hatred to make both the Uchiha and Hyuga clans like yin and yang. The Uchiha become stronger through negative emotions, while the Hyuga become stronger through positive emotions.
As the Mangekyo Sharingan was different for different users, we will give the Bankakyo Byakugan a color based on their chakra:
For Neji, it will be green.
Violet for Hinata.
Teal for Hanabi.
Cyan for Toneri.
Blue for Hiashi.
Purple for Hanashi.
Emerald for Hizashi.
As for the abilities:
Hizashi's Bankakyo Byakugan: The left eye will give foresight like Neji's, but for 5 seconds. The right eye will give instant teleportation within the field of vision (Itachi-Sasuke connection between Hizashi and Hiashi).
Hiashi's Bankakyo Byakugan: The right eye will give instant teleportation within the field of vision but only on yourself (same as Hanabi's because of the parent-child connection). The left eye will give chakra-formed armor (same ability as Hinata because of the parent-child connection).
Neji's Bankakyo Byakugan Kuninotokotachi: Left eye can see the future for 10 seconds (foresight).
Kuninotokotachi: Right eye can skip time (like Hit’s time skip from Dragon Ball Super), improving his taijutsu. You can only skip time that you can see. Death is unavoidable even with this ability. Damage done to the body doesn't go away unless you can heal it within 10 seconds.
Hinata's Bankakyo Byakugan: The left eye will give her chakra-formed armor for greater defense or attack. The right eye will give her an air palm-like attack but from any chakra point in her body. She can use it to make a shield.
Hanabi's Bankakyo Byakugan: The left eye will give her an air palm-like attack but from any chakra point in her body (similar to Hinata; Itachi and Sasuke have similar abilities). The right eye will give her teleportation within her field of vision in a 20-meter radius (like Goku's instant transmission), usable on anyone in the field of vision.
Hanashi's Bankakyo Byakugan: The left eye will give her an air palm-like attack from any chakra point in her body (like her daughter Hanabi). The right eye will give her chakra-formed armor for greater defense or attack (like her daughter Hinata).
Toneri's Bankakyo Byakugan: The left eye will give him control over the body of others for 10 seconds. The right eye will give him the ability to absorb chakra by touching or through balls like in the last movie.

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