If you liked
Tokyo Godfathers
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...then you might like
NHK ni Youkoso!
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Both Tokyo Godfathers and Welcome to the NHK! have hilarious characters while also having a pretty serious overarching plot that shows how life can beat you down. While TG is more serious for the most part, NHK focuses more on comedy. However, it has its fair share of melancholic episodes too, especially towards the end. The color schemes are similar, with a lot of gray and black to emphasize the feeling of loneliness. These are both anime classics, and if you like one of them I'd have to imagine you will like the other.
If you liked
Koukaku Kidoutai
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...then you might like
Psycho-Pass
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Both of these anime have similar settings, and feature a smart female detective as the leading protagonist. They have similar themes about the problems prevalent in a society which relies too heavily on technology. After all, there are always bugs or loopholes that can be taken advantage of. The art and sound in both are brilliant as well. Finally, the most notable similarity are the heavy philosophical themes in both shows. GitS and PP both analyze human society, and also try to define humanity itself.
If you liked
Monster
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...then you might like
Psycho-Pass
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Both of these shows involve catching a very smart and charismatic criminal. Both shows also delve into big societal issues and the underlying shortcomings of human nature. Psycho-Pass is much faster paced overall, and has more action, while Monster progresses slowly and likes to focus on its many side characters and side stories, with huge payoffs later on. Both shows also have terrific voice acting (both sub and dub) and extremely well-written dialogue.
If you liked
Tonari no Totoro
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...then you might like
Ookami Kodomo no Ame to Yuki
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Both of these movies have similar themes of family and growing up. Totoro is more geared towards kids while Wolf Children appeals to adults more (specifically parents). This also means that Totoro is more lighthearted and fun while Wolf Children is somewhat on the serious side, although it still has plenty of funny/cute moments. These two movies also have similar settings, featuring a parent with two kids in a house near the forest. The soundtracks are very different, but both sound great and really help boost the impact of important scenes. I would also say that these two movies are the best animated movies of their respective decades.
If you liked
Rurouni Kenshin: Meiji Kenkaku Romantan
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...then you might like
Fullmetal Alchemist: Brotherhood
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Rurouni Kenshin and FMA: Brotherhood may have entirely different settings and stories, but the pace and character development in these two shows could not be more similar. Both shows strongly emphasize character development, fully developing the main cast as well as the side characters and villains so the viewer can fully understand all their perspectives. Also, the soundtracks of both shows are flawless.
If you liked
Fate/Zero
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...then you might like
Sword Art Online
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The soundtracks for SAO and Fate/Zero sound very similar. Both are filled with great dramatic themes as well as light sad tunes. They are both of course composed by the legendary Yuki Kajiura. The two shows also have equally superb action sequences and animation.
If you liked
Cowboy Bebop
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...then you might like
Baccano!
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Baccano! and Cowboy Bebop have a very similar cool and jazzy atmosphere. The music and intros of both shows are also similar; both intros are swingy jazz tunes. Both these shows are specifically great for western audiences, and they both have superb English dubs.