If you liked
Zoku Natsume Yuujinchou
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...then you might like
Fruits Basket 1st Season
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-Both of them are heartwarming animes that can be sad at times - Both Mc's are selfless & kind people - Both are slice of life drama's that deal with somewhat supernatural things (one being yokai and the other being zodiac signs) - Soundtracks are similar
If you liked
Shigatsu wa Kimi no Uso
|
...then you might like
Plastic Memories
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Plastic Memories is a romantic anime with a sadish twist to it just like Your lie in April. Plastic memories is revolved around robots and memories while with Your lie in April its more revolved around music and friendship. My experience with plastic memories is that youll most likely cry in more than one episode which may happen if you have watched Your lie in April. Both of these animes can leave an ever lasting impression on you and they both have beautiful opening and ending songs. Both of these animes are sad though so id say watch at your own risk.
If you liked
Elfen Lied
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...then you might like
Mahou Shoujo Madoka★Magica
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Elfin Lied is another dark anime but its "darkness" is shown much earlier (first few seconds instead of the first few episodes). I'd definitely say that Elfin Lied has a darker concept and is wayyyy more darker than Magica Madoka. The bad thing is that Elfin Lied doesn't have the cutesy artstyle that Madoka has, buttt I'm sure if you enjoyed the dark theme that Madoka Magica had then I'm sure you'll enjoy Elfin Lied a lot.