I recently asked: Completion of $S\cong \Bbb Q$.
Note: If the commenters and answerers could include (why is this question good and why is it bad) I would be grateful for that.
What do the experts think about this question? (I would especially like to hear from Peter Scholze).
I think I should have taken a few more minutes to collect my thoughts before writing. However I think there is some research level material in this post.