It is proven in SGA1 that if $k$ is an algebraically closed field, if $X$ is a proper $k$-scheme and if $Y$ is a locally noetherian $k$-scheme (say, $X$ and $Y$ are non-empty and connected) then $\pi_1(X\times_k Y)=\pi_1(X)\times \pi_1(Y)$. This implies that if $L$ is any algebraically closed extension of $k$ then $\pi_1(X\times_k L)=\pi_1(X)$. All of these results are known to fail without the properness assumption, but all counter-examples are in positive characteristic, and I am almost sure that this assumption can be removed in char. $0$.
Question: is it written somewhere? I did not find any proof of it in SGA, nor in Stacks Project.
For the commutation with the product over $\mathbb C$ one can use the corresponding result in topology and the comparison theorem between étale and topological fundamental groups, but reducing the case of an arbitrary algebraically closed field of char. 0 to that of $\mathbb C$ would first require a proof of the invariance of the $\pi_1$ under algebraically closed extension.