Netflix removed every Takashi Miike film aside from Oldboy. >.> If you're bored and have a way to check out non-Netflix movies, just go through his library of films. Audition was crazy, he did a short for one of the Three Extremes movies that was great (all of the shorts in the movie are really good; Dumplings also got made in to a full movie), Ichi the Killer was off the damn wall, and Imprint was one of my favorites of his. Also Park Chan-wook is another incredible director with some great films to his name. The Vengeance trilogy is really good. He's also the director of Oldboy (which is part of that trilogy). I wasn't a huge fan of Snowpiercer, but it has some good actors (Song Kang-ho from the Vengeance trilogy and The Host, and Tilda Swinton) that made it entertaining for me to watch.
Invisible Secret was on Netflix, but they removed it.
Watched that flick on a whim one night and fell in love. Some other awesome non-Netflix movies I'd highly recommend are A Tale of Two Sisters and Carved/The Slit-Mouthed Woman. Carved is beyond creepy, super unnerving and atmospheric. Kim Jee-woon, who directed Tale, also directed I Saw the Devil which was great. Battle Royale is what The Hunger Games wishes it was. If you like THG or the idea of it, or if you hated THG and wished it was better, Battle Royale completely blows it out of the water. There's a movie called Kwaidan from the '60s that is a little difficult to find, but seriously worth it if you come across it. Something like 3 or 4 short stories, really really good stuff. Marebito is a freaky supernatural movie, worth a watch if you come across it somewhere. I remember Apartment 1303 being a pretty good paranormal/ghost flick too.
Also if you want to check out a darker anime, I very highly suggest you watch Knights of Sidonia on Netflix. Big monsters and big robots, all in outer space. Attack on Titan is another darker anime, although not on Netflix, but one of the best shows I've ever watched. A population of people have been living inside a walled city to protect them from Titans, which are gigantic human-looking things that eat people for seemingly no reason and the show has revolved around an elite fighting squad tasked with protecting the city/gathering information about what they are. Second season recently started too.
I've been meaning to check out this movie called Memento Mori. It's part of a series of films under the Whispering Corridors name that seem to get a lot of positive reviews. Suicide Club and Noriko's Dinner Table have also been on my to-watch list for years now. Reading about it, Gozu sounds really good and I have to add that to my list.