I will make a controversial choice of Lars Von Trier. He's just on the mainstream side of 'arthouse' and his films are incredibly divisive, but I find something interesting and valuable in all of them. I'd recommend Melancholia as a starter, then Dogville or Dancer in the Dark.
Cristian Mungiu, part of the Romanian New Wave, whose films are some of the most beautifully modern and thematically sophisticated being produced at the moment. I'd recommend 4 Months, 3 Weeks and 2 Days and Beyond The Hills, with 4 Months perhaps being the first watch.
Krzysztof Kieślowski, whose films border on realist fantasy with the way timelines appear to cross and characters appear to blend into one another whilst still remaining earnestly grounded. I'd recommend the Three Colors Trilogy, which really need to be watched the order they were intended.
David Lynch, who is decently well known but is still under appreciated, a genuine master of the uncanny whose films are creepy, unsettling but always unique and frequently filled with pathos and poignancy. I'd recommend Blue Velvet as a starter, then move onto Mulholland Drive and Eraserhead.
Park Chan-wook, the best visual artist of the past twenty years in terms of cinema, with shots so unbelievably well composed and intricate you could frame almost anyone of them. His storytelling is a twist on traditional tragedy, mixing darker humour elements before pulling the rug out from under you with climax's that ache with sadness. I'd recommend JSA to start, then move onto the Vengeance Trilogy.
Jean Luc-Godard, who has recently gone insane and produced nothing but gaudy trash, but Breathless is one of the most vital and important works of cinematic history. The man completely rethought cinematic space and editing and gave us the sorts of films we watch today.
There are a few others I can mention, but I think I'll stop there.
