 If you liked
Higurashi no Naku Koro ni Kai
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 ...then you might like
Mahou Shoujo Madoka★Magica
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Probably a huge spoiler, but their point of origin is exactly the same. Mahou Shoujo Madoka★Magica seemed like the "sugar-pink contacts on for some tension" version of Higurashi series after I finished watching it.
Both animes try to change fate by retrial, both are very psychological, making you think about human values.
Kai and the last few episodes of Madoka Magica both follow a similar story of changing fate
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Recommended by Rinthronhael
Both of these series have a similar style, both have this naive design that may look like it's just childish anime even when they are so dark and deep. Not only this, but those two also have similar plotlines when it comes to the development of the series. Both are really great at character and plot development. With this said, I could not recommend a better anime that looks like Higurashi.
I was originally going to write yet another recommendation for Steins;Gate, but after a while I decided on recommending Higurashi Kai to PMMM instead. Here's why~
Well, both shows obviously use time travel as a major point of the plot. You can expect to see a character endlessly repeating the same sequence of events over and over again to attain a desired outcome in both.
However, the reason why I didn't just point you guys to the ultimate granddaddy of time travel anime, Steins;Gate, was because both Higurashi Kai and PMMM have a "mysterious" feeling to them. Both give you an uneasy feeling, which is supported by
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**POSIBLE SPOILER**
Time travelling, an dark psychotic atmosphere, the main protagonist suffering like a bitch and a lot of deaths (Higurashi's are goriest). Is easy to see their similities cause Rika and Homura are practically the same character, both having to watch their beloved ones diying, having a very serious personality 'cause the timetravels, watch their friends going crazy in some timelines and both are a hidden protagonist. Although the reason why Rika and Homura go through time are different, the two have to be with saving another people.
Also, the two are masterpieces and this is practically demostrated with their high rating in MAL!
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Recommended by SuzuMine-chan