Let $X$ be a smooth projective variety over a field $k$. By (one) definition, the Chow group of zero-cycles $CH_0(X)$ is universally trivial if, for every field extension $k \subset K$, the degree map $CH_0(X_K) \rightarrow \mathbf Z$ is an isomorphism.
This idea has been an important ingredient in some major recent advances in birational geometry. In particular, B. Totaro showed ("Hypersurfaces that are not stably rational", available here) that if $X$ is a very general hypersurface of degree $d$ in $\mathbf P^{n+1}_\mathbf C$ and $d \geq 2 \lceil (n+2)/3 \rceil$, then $CH_0(X)$ is not universally trivial, and hence $X$ is not stably rational.
Of course, the significant cases are when $X$ is Fano, that is, when $d \leq n+1$. Here $X$ is rationally connected, so $CH_0(X) \cong \mathbf Z$; Totaro's theorem implies that after some base extension $\mathbf C \hookrightarrow F$, the Chow group $CH_0(X_F)$ is bigger than $\mathbf Z$.
Question: For $X$ and $d$ as above, what kind of field extension $\mathbf C \hookrightarrow F$ makes $CH_0(X_F)$ bigger than $\mathbf Z$?