# Why is $\mathbb{C}$ over $\mathbb{R}$ considered ramified?

For a number field $$K/\mathbb{Q}$$, we say that a finite place of $$Q$$ is ramified if there exists a valuation $$v_{p_i}$$ in $$K$$ lying over $$v_p$$ such that it is ramified in the sense of the associated discrete valuation rings. This is the ramification index, denoted $$e_{p_i/p}$$.

We also have the residue field extension, of degree $$f_{p_i/p}$$.

With these, we can define $$n_{p_i/p}$$ as $$f_{p_i/p}e_{p_i/p}$$, and the $$n_{p_i/p}$$ behave well, they sum to the degree of the extension and so on.

Now this is all lovely over a finite place, but for infinite places, in the treatment I saw, only the integers $$n_{\infty_i/\infty}$$ were defined, as $$1$$ if the associated extension is real to real, and $$2$$ if real to complex. These infinite $$n_{\infty_i/\infty}$$ behave in the same way as the finite $$n_{p_i/p}$$.

So then the question is, why do we view $$n_{\infty_i/\infty}=2$$ as ramified, eg, from the perspective of CFT. Is there an intuitive explanation of why these ought to be ramified and not just the analogue of a purely $$f_{p_i/p}$$ extension of finite places?