If you liked
Arakawa Under the Bridge
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...then you might like
Danshi Koukousei no Nichijou
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Both are over the top, absurdist comedies. Danshiko is the funniest anime I have ever watched, one that manages to make me and my friends laugh even during rewatches. It doesn't have much plot if any, unlike Arkawa. Arakawa's humor kind of wore thin for me, whereas Danshiko kept me laughing for its entirety. Danshiko's strong suit was how it was so funny while still being grounded in reality (i've seen some the scenarios in Danshiko being played out in real life). Both are anime that you probably wouldn't regret watching (I prefer Danshiko), but if you just want to really laugh, watch Danshiko, if you want something kinda funny with plot (its humor is really a hit or a miss), watch Arakawa.
If you liked
Kanon (2006)
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...then you might like
Clannad
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Factual similarities: Both are based on visual novels produced by Key, both are harem and have 5 heroines, and both have a conclusive ending. My opinion: Kanon is much wider and cohesive in its focus, it gives importance to most of the main characters compared to Clannad. Clannad has a much greater emphasis on family; in Kanon you don't know the protagonist's family, whereas in Clannad, family (not just of the protagonist) plays a crucial role. Kanon is more about lost connections. Clannad is definitely more plot driven compared to Kanon. I also think the humor flows better in Kanon than in Clannad, and the protagonist in general is funnier. Both are lacking in the character design area with their bug eyed heroines, but you get used to it. Both are a must watch.
If you liked
Shinseiki Evangelion
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...then you might like
Mahou Shoujo Madoka★Magica
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Both anime can be seen as successful deconstructions of their respective genres. Both involve people fighting against an enemy they don't really understand. NGE is darker than Madoka. NGE is also harder to interpret, and requires at least a second viewing to really have a idea of what is going on. Both are definitely worth watching, but recommended only after you have some experience with anime. Definitely not recommended as someone's first anime. A significant part of the appeal of these anime is how they compare to shows similar to them.
If you liked
Mahou Shoujo Madoka★Magica
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...then you might like
Gakkougurashi!
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Factual similarities: Both anime are about young girls. Both anime try to do something very different with their respective genres. My opinion: Madoka seems to succeed in deconstructing its genre, unlike Gakkougurashi!. I think Gakkougurashi! didn't go far enough, it spent too much time in the middle, unsure of what it wanted to be. I think Madoka Magica was by far the better of these two.
If you liked
Shinseiki Evangelion
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...then you might like
NHK ni Youkoso!
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Both are psychological anime that deal with depression. The main character in both have a strong desire to 'run away' from the problems facing them. Neon Genesis Evangelion is considerably harder to digest than NHK. It is recommended both be watched without reading to much about them, both are good, just go watch them.
If you liked
Shuffle!
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...then you might like
Kanon (2006)
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Factual similarities: Both are harems based on visual novels. Both have conclusive endings. Both have supernatural elements. My opinion: Kanon is much better than Shuffle! though, much much better. I disliked Shuffle!; the characters have no variety and have extremely similar origin stories, the protagonist is awfully boring, a world that is not really explored at all. Kanon has most of the things Shuffle! lacks.
If you liked
Arakawa Under the Bridge
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...then you might like
D-Frag!
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The type of comedy is kind of similar, both protagonists are the 'straight men' surrounded by weird and eccentric characters who aren't aware of how weird they are. Arakawa Under the Bridge has more absurdest comedy than D-Frag, to the extent where 'mentally ill' might be a better term for most of the characters than 'eccentric'. Arakawa could sometimes get slow at parts, that wasn't an issue with D-Frag (but D-frag is shorter). Neither anime has a conclusive ending. D-frag is in highschool, Arakawa's cast is mostly adult.
If you liked
Ano Hi Mita Hana no Namae wo Bokutachi wa Mada Shiranai.
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...then you might like
Kokoro Connect
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Factual similarities: Both are high school anime with romance and supernatural elements. The supernatural elements in both are used to create drama. My opinion: I didn't like either of these, however, of the two, I think Anohana might be the less worse one. The drama in Anohana was somewhat justifiable, not so in KokoroC. KokoroC dramatizes something almost everyone goes through, without making such a fuss about it. In Anohana, a lot of the characters sometime start crying and start shouting out their feelings instead of the reader letting them figure them out. That was just bad writing. KokorC has an issue with inconsistent characters. Anohana's focal character was plain annoying, and could do something which would have fixed many issues if done from episode 1.
If you liked
NHK ni Youkoso!
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...then you might like
Yojouhan Shinwa Taikei
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Both follow adult characters discontent with their lives , and their coming to terms with their issues. Both make use of visual storytelling, though Takei does it more often and more explicitly. Takei is a bit more philosophical, and perhaps harder to grasp than NHK. I really like how they take a more realistic approach to the characters' issues instead of a more idealistic one.
If you liked
Kenpuu Denki Berserk
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...then you might like
Hellsing Ultimate
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Both have a seemingly unkillable main character indulging in gratuitous violence, and philosophical undertones (which I think Berserk does better).
If you liked
Hellsing Ultimate
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...then you might like
Black Lagoon
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Both have amoral, almost invincible, dual gun wielding main characters who are kind of anchored by another, relatively less screwed up main character. Also, guns. Lots and lots of guns. And explosions. And guns. Watch these shows for the action, they really knock it out of the park in that department. I myself prefer Hellsing Ultimate because its supernatural elements provide for better entertainment value in terms of action.
If you liked
Claymore
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...then you might like
Black Bullet
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Factual similarities: Both anime have humanity fighting monsters by using monster-human hybrids (all of whom happen to be female) who are at a constant risk of becoming monsters themselves. Discrimination against them is a common theme. The main focus is on a pair consisting of a human male and one of the aforementioned hybrids. My opinion: I didn't like either show, but I think Claymore is the less worse of the two. The only major issue with Claymore is really just one character. Black Bullet, on the other hand, has a whole host of problems. BB is one of the more generic shows I've watched; I can't put my finger on what exactly made me dislike it because there were so many reasons.
If you liked
Darker than Black: Kuro no Keiyakusha
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...then you might like
Dimension W
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Darker than Black is essentially the better version of Dimension W. The character designer for DTB is the writer of Dimension W, the protagonist is a taciturn bad ass who fights using knives and ropes, with a troubled past, and other similarities that would spoil the story. Seriously, just watch DTB.
If you liked
Seto no Hanayome
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...then you might like
Arakawa Under the Bridge
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Romantic anime with extremely similar style of comedy. Seto no Hanayome's humor isn't as nonsensical, has a greater emphasis on romance, and has a proper ending. I prefer Seto over Arakwa, because the ending is paramount. Arakawa's humor can be so over the top at times that you start to tire of it.
If you liked
Higurashi no Naku Koro ni
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...then you might like
Denpa-teki na Kanojo
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Both mislead the viewer initially, only to take some very interesting turns later on. For the best experience, watch without reading the plot or reviews. The vibe and the feel they give off is very interesting, and with both shows it is hard to pinpoint what exactly makes them good.