Izuku's Quirk is similar to DC Comics Doomsday (his quirk name is One Wrath) that allows him to create a new power after dying (again) or being in a overly stressful situation. Also when he dies the first time whoever he hates the most he'll have a power similar to that person's quirk, which in this case it will be shared by All Might and Inko due to their Neglectful (Inko) and Abusive (All Might) Treatment, and as such at first he'll have the power of his canon version but the difference will be that how much power he has will be based on his bodies endurance and durability rather than his physical capabilities and the lightning effect is a side effect of his hatred for Inko. His hatred for Inko gives him telekinetic tendrils which manifest as erratic green electricity.
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Wrath Of The End
FanfictionIzuku Yagi has a quirk, but everyone believed (Himself Included) that he didn't. While his parents and the world turns on him, very few stay by his side... This is a somewhat OP Neglected Izuku story but beyond that it throws a majority of all those...