Unprovability is a property of a statement given certain axioms / a theory (assuming that it can be formulated in this theory). Classical examples include the unprovability of the Parallel Postulate over Euclid's other axioms and the unprovability of the Continuum Hypothesis over ZF. Concepts like Forcing and Gödel's incompleteness theorem are popular tools in this area.
Are there known statements which are unprovable over $\mathbb{R}$ (defined axiomatically), but provable over $\mathbb{C}$ and vice versa? Bonus question: Can we even find examples for statements unprovable over a field $K$ but provable over a nontrivial field extension $K(x)$?
The question arose purely by accident, and I would be grateful too for hints regarding literature that treats this topic.