The étale fundamental group changes when the base field changes

A finite étale algebra over $\mathbb{C}$ is a finite product of copies of $\mathbb{C}$ by [F1] and the finite-étale geometric-fibre assertion in [F2]. Its fibre functor is therefore the usual finite-set functor on disjoint unions of the basepoint. A natural automorphism of this functor fixes the singleton fibre of the identity cover, and by naturality for all maps from that singleton it fixes every point of every finite fibre. Hence ${\pi }_1^{\mathrm{e}\mathrm{t}}(\mathrm{Spec}\mathbb{C})=1$.

The algebra $\mathbb{C}=\mathbb{R}[T]\mathrm{/}(T^2+1)$ is free of rank two over $\mathbb{R}$, and $2T$ is invertible in it, so it is finite étale by [F2]. Its spectrum is connected. Its two geometric points over the chosen complex basepoint correspond to the embeddings sending $T$ to $i$ and to $-i$. Complex conjugation interchanges them; it is the unique nonidentity deck transformation, since an automorphism is determined by its action on the image of $T$. Thus the cover is Galois of order two. By [F2], ${\pi }_1^{\mathrm{e}\mathrm{t}}(\mathrm{Spec}\mathbb{R})$ surjects onto that deck group, and cannot be trivial. This differs from step 1.1 after the stated base change and refutes the claim.