Irotoridori No Sekai - The Colorful World

Our story follows Kanoue Yuuma, an amnesiac boy with a special power, and Nikaidou Shinku, a translucent girl only visible to Yuuma who gave him that power. With it, he can heal any and all wounds. But it comes at the cost of the memories he's made. He lives a simple life: going to school and being scolded by his childhood friend, Kisaragi Mio, by day, and working with his master, Natsume Suzu, in the evenings. However, whenever he falls asleep, he always sees the same dream... a white world, feathers falling, and a girl calling out for him to save her. The student dormitory his master resides in has a special door in the basement, which connects their world to a myriad of different parallel ones. He travels from world to world, searching for one that resembles the scenery he sees in his dreams, but to no avail.

Then, one night, as he's out at night strolling around town, he spots a girl jumping from the lighthouse. Yuuma rushes over and heals her using his power, but after coming to, the girl claims to know him and his power, and asks him much like the voice in his dreams:

"Please save me, Kanoue Yuuma."

[From (https://store.steampowered.com/app/2591380/Irotoridori_No_Sekai_HD__The_Colorful_World/)]