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She had initially been Martha Wayne, the mother of Bruce Wayne, who would have become Batman. Unfortunately, a twist of fate occurred after Barry Allen, aka the Flash, changed the timestream, which resulted in her and her husband Thomas Wayne surviving Joe Chill's fateful ambush, but their son dying as a result. This ultimately broke her, driving her completely insane from grief and having her adopt the identity of the Joker, as well as committing various crimes in Gotham, and also resulting in Thomas becoming a violent vigilante known as Batman in an attempt to stop her.
By the time of the Flashpoint saga, she had kidnapped the twin sons of Harvey Dent without leaving a ransom note. She then, at her hideout of Wayne Manor, manipulated Commissioner Gordon into shooting the daughter by dressing her up in a Joker outfit, and then murdered the Commissioner. Eventually, Batman arrived, and she proceeded to bludgeon her former husband with a hammer, an action Batman willingly endured for a chance at getting a glimpse of the remaining psyche of his former wife. Ultimately, he managed to temporarily get through to her, and revealed that they may have a chance of undoing the events that happened, although it may result in their deaths. Martha accepted, although upon learning what will become of Bruce (that he'll become Batman, like his father), she goes insane again and commits suicide.
Martha goes insane.
Martha Wayne/Joker in Justice League: The Flashpoint Paradox.
On a related note, this version alongside the Nolanverse Joker and Arthur Fleck are the only Joker incarnations who hadn't been dunked into a chemical vat beforehand as part of their becoming the Clown Prince of Crime.
Ironically, this is also the only version of the Joker whose origin story stemmed from the exact same event as Batman's origin story (in this case, the death of Bruce Wayne).
This is one of only two villainous Joker incarnations to be female instead of male, the only other being Bianca Steeplechase.
Martha Wayne is one of the three Jokers who have their real name revealed, the other two being Jack Napier and Arthur Fleck.
Given that Martha Wayne is the Joker in the Flashpoint timeline, it's beyond anybody's guess what became of the man who became the Joker in such timeline. However, if we consider Batman: The Killing Joke as the official backstory of the Joker as heavily implied, we can deduce that as Batman perhaps never came there due not being Bruce Wayne, the unstransformed Joker was either shot by the police, arrested or managed to get away, while other possibility could be that Batman (Thomas Wayne) went there instead and both the untransformed Joker along with Vinnie & Joe were killed by him.
This is one of the few Joker incarnations to be portrayed as a truly sympathetic character, with another being the version from the animated adaptation of Batman: The Killing Joke. Other examples would be the version from Batman: The Telltale Series and the version from the eponymous 2019 film.