Is there a solution to the green shadows problem?

Beleive it or not, it's an artistic decision. I actually like it personally.

Though I think it's to cover up for how the game's specific texture compression (JXR aka. Microsoft HD Photo aka. JPEG XR) doesn't work as well for dark colours as it does for other colours.

I've tried the image format myself with XnView (free tool to convert images to tons of different formats) and it works out very well and produces great quality at low filesizes, only problem is there is some pretty evident compression artifacts with the dark colours.

if ID decided in this way, no problem, I'm enjoying the game but I would like to enjoy the pure experience without glitches or problems.
If you say that the shadows are green in this game, I'm fine.

Well the shadows outside should be 'normal dark', but I take it you are referring to the green-tinted shadows inside caves right?
Don't worry then. They also were like that in the official videos.

yes I mean the shades in the caves, if they looked green in the official videos, great, no problem, thanks for the answer.