Off the top of my head, I'd say:
1) Death Note: Was pretty intense. Very psychological and asks a lot of questions on morality.
2) Mirai Nikki: Same as Death Note, but includes romance.
3) Deadman Wonderland: Similar to the above, but includes a sort of super power.
4) High school of the dead: Same as above, but with ecchi and romance.
All those anime have plenty of action and will have you asking yourself questions on morality. I recommend them all!
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