So, the plot of Bayonetta 3 kicks off with some undefined gribbly killing Bayonetta in the Game 1 costume.
This is Singularity. A literal fucking AI that Skynet'd his original reality and discovered the existence of parallel worlds, he decided the only reasonable thing to do was to go multiverse Highlander and destroy other realities to try and seize control of the full power of the World of Chaos. To that end, he's been hunting down and killing Bayonettas, calling her "Arch-Eve."
The Bayonetta we're playing is enjoying a peaceful day and is then promptly jumped by Singularity's Homunculi army, who notably are not in Purgatorio. There's a big fight and when Bayonetta summons Gommorah out of the dog house to finish some off, he refuses because he's a picky fucking eater and only wants Angel flesh. Bayonetta, utterly tired of his shit, uses a spell called Demon Slave to force Gommorah to eat anyway.
After a big flashy battle, Bayonetta is in Rodin's shop to work out what to do about these yahoos, at which point Violet shows up. Violet serves as exposition dump and explains basically what they are, what they want, and what to do about them(track down some Umbran/Luman artifacts called Chaos Gears to be able to go directly to the "Alphaverse" rather than try and hoof it through Singularity's controlled realities.
Bayonetta and Violet go to the island of Thule that the Witches and Sages made to study parallel realities while Jeanne goes looking for a "Dr. Sigurd" who Violet says is important.
While on Thule, Bayonetta goes her own way and tells Violet to go keep an eye on Luka(who also showed up as he does) while she hunts down Chaos Gears.
Bayonetta proceeds to go through a variety of levels in parallel worlds and meets alternate versions of herself(Bayonetta in Tokyo, Ancient Chinese Bayonetta, Egyptian Princess Bayonetta(my personal favorite), and French Phantom Thief Bayonetta, all of whom fucking die in the course of the story.
While fighting Singularity and his goons across the multiverse, Bayonetta is forced to use the Deadly Sin ritual several times, which involves ripping her heart out to super charge her demons.
While this is happening, Violet tracks down Luka who is turning into a werewolf for unknown reasons. Eventually she meets an alternate Luka called "Lukaon" who is a faerie king. When Violet is about to be killed by Werewolf Luka, she basically Devil Triggers.
Meanwhile, Jeanne is fucking around in a spy minigame trying to find Dr. Sigurd and eventually gets him and drives off into the sunset.
The three meet up and try Dr. Sigurd uses the Chaos Gears and Thule to open a path directly to the Alphaverse so Bayonetta can throat kick this problem. Her, Violet, and Luka(who has gained control of his werewolf form) get going while Jeanne and Dr. Sigurd stay behind to make sure that Singularity doesn't blow up their portal.
There's a flashy boss battle and it turns out the Singularity they thought they were fighting wasn't actually Singularity. Dr. Sigurd turns out to be Singularity and then kills Jeanne.
Bayonetta and co are trapped in the Alphaverse, but Lukaon shows up and helps them out revealing that Luka is the Arch-Adam to Bayonetta's Arch-Eve.
Bayonetta gets back to her reality and fights Singularity in a big flashy final boss battle. During which time, she punches him so hard the quantum ghosts or souls of all of the dead Bayonetta's(and Jeanne) show up to help beat the shit out Singularity. This goes on for a while, then Singularity gets fed up of the violent gangbang and banishes/kills them all. Regular Bayonetta keeps fighting him and eventually the Bayonetta's from the First and Second game show up to help. They beat the shit out Singularity for a hot minute, do some kind of fusion dance, beat the shit out of him some more, and eventually the extras get got and regular Bayonetta is running on fumes.
Because while Singularity may not have been quite the foe that Jubileus or Loptr were, he graduated with top marks at the Freeza school of "This isn't even my final form!"
Violet and Luka show up to help and Violet is revealed to have been (an alternate) Bayonetta and Luka's daughter. A fact that was transparently obvious to fucking everyone long before this, thankfully including Bayonetta and Luka. Bayonetta pulls out one last Deadly Sin, summons Gommorah and finally beats Singularity.
Unfortunately, while Singularity collapses into a black hole, Gommorah slips his leash AGAIN and prepares to kill Bayonetta, who's basically exhausted and defenseless. Meanwhile Violet is getting sucked into Singularity's death spasm and Luka agonizes before saving Violet while Gommorah claws Bayonetta's soul out of her body.
Luka gets back in time to have a moment with Bayonetta before she's dragged to hell and he joins her.
Violet then gets jumped by the collected dark emotions and such of all the murdered Bayonettas of the multiverse and has a last fight with Dark Eve Bayonetta. She wins and resolves all of the negative emotions of the Bayonettas, who has a moment with her, tells Violet she's a grown up Umbra Witch and gives her the name "Bayonetta."
At the epilogue, Violet is shown doing jobs for Rodin, who makes a comment about how if she isn't also doing well in school he'll catch an earful from her parents. Most fans have decided this means Luka and Bayonetta aren't "Wandering scared shitless for eternity" as the first game stated, but have gotten a position of power and become demon lords in Inferno.
Which, to be fair, there are lore entries of people who did exactly that, notably Madama Butterfly. So, not exactly wishful thinking.
