3 Incorporated comment

I can only offer a "strengthening" of your friends' explanation. Let me first remark that I am not an expert in this field and I am sure that there are some grave mistakes in my argument. However, it is much too long for a comment, so I post it as an answer.

Let us first consider the simpler case of (co)homology instead of fundamental groups. When talking about étale (say, $\ell$-adic) cohomology together with its Galois action, the transcendental analogue is generally taken to be not just the singular cohomology groups, but these groups together with their Hodge structure.

Similarly, consider a hyperbolic curve $X$ over a number field $K$. For simplicity, assume we are given a $K$-rational base point $x\in X(K)$. The fundamental group one considers is either the group $\pi_1^{et}(X,x)$ as an abstract profinite group, or the group $\pi_1^{et}(X\otimes\overline{K},x)$ together with its action of $\operatorname{Gal}(\overline{K}|K)$. The former can be reconstructed from the latter. The weakest version of Grothendieck's anabelian conjecture for curves says (roughly) that we can reconstruct $X$ from $\pi_1^{et}(X,x)$.

Let me explain why we can reconstruct $X$ from $\pi_1^{et}(X\otimes\overline{K},x)$ with its Galois action. The abelianization of this group with Galois action is just the product over all $\ell$ of the $\ell$-adic Tate modules $T_{\ell }(\operatorname{Jac}X)$. These are the $\ell$-adic analogues of the Hodge structures on first homomology, which bear the same information as the Jacobian itself. Thus it is not surprising (although very difficult!) that we can reconstruct $\operatorname{Jac}X$ from these data, and the Jacobian determines the isomorphism class of the curve by Torelli's theorem. [Edit: As Torsten Ekedahl has pointed out in the comments, it is not true that you can recover an abelian variety from its Tate module.]

Now there are certainly some points where the above argument does not work as simply as presented, but the morals is that the analogue of the arithmetic fundamental group over $\mathbb{C}$ should be the topological fundamental group with a "Hodge structure on groups". I do not know if this has ever been worked out, but there is a good understanding of the "Hodge structure on the nilpotent completion of the fundamental group", introduced by Hain and Zucker.

2 formatting improved

I can only offer a "strengthening" of your friends' explanation. Let us first consider the simpler case of (co)homology instead of fundamental groups. When talking about étale (say, $\ell$-adic) cohomology together with its Galois action, the transcendental analogue is generally taken to be not just the singular cohomology groups, but these groups together with their Hodge structure.

Similarly, consider a hyperbolic curve $X$ over a number field $K$. For simplicity, assume we are given a $K$-rational base point $x\in X(K)$. The fundamental group one considers is either the group $\pi_1^{et}(X,x)$ as an abstract profinite group, or the group $\pi_1^{et}(X\otimes\overline{K},x)$ together with its action of $\operatorname{Gal}(\overline{K}|K)$. The former can be reconstructed from the latter. The weakest version of Grothendieck's anabelian conjecture for curves says (roughly) that we can reconstruct $X$ from $\pi_1^{et}(X,x)$.

Let me explain why we can reconstruct $X$ from $\pi_1^{et}(X\otimes\overline{K},x)$ with its Galois action. The abelianization of this group with Galois action is just the product over all $\ell$ of the $\ell$-adic Tate modules $T_{\ell }(\operatorname{Jac}X)$. These are the $\ell$-adic analogues of the Hodge structures on first homomology, which bear the same information as the Jacobian itself. Thus it is not surprising (although very difficult!) that we can reconstruct $\operatorname{Jac}X$ from these data, and the Jacobian determines the isomorphism class of the curve by Torelli's theorem.

Now there are certainly some points where the above argument does not work as simply as presented, but the morals is that the analogue of the arithmetic fundamental group over $\bbc$ \mathbb{C}$should be the topological fundamental group with a "Hodge structure on groups". I do not know if this has ever been worked out, but there is a good understanding of the "Hodge structure on the nilpotent completion of the fundamental group", introduced by Hain and Zucker. 1 I can only offer a "strengthening" of your friends' explanation. Let us first consider the simpler case of (co)homology instead of fundamental groups. When talking about étale (say,$\ell$-adic) cohomology together with its Galois action, the transcendental analogue is generally taken to be not just the singular cohomology groups, but these groups together with their Hodge structure. Similarly, consider a hyperbolic curve$X$over a number field$K$. For simplicity, assume we are given a$K$-rational base point$x\in X(K)$. The fundamental group one considers is either the group$\pi_1^{et}(X,x)$as an abstract profinite group, or the group$\pi_1^{et}(X\otimes\overline{K},x)$together with its action of$\operatorname{Gal}(\overline{K}|K)$. The former can be reconstructed from the latter. The weakest version of Grothendieck's anabelian conjecture for curves says (roughly) that we can reconstruct$X$from$\pi_1^{et}(X,x)$. Let me explain why we can reconstruct$X$from$\pi_1^{et}(X\otimes\overline{K},x)$with its Galois action. The abelianization of this group with Galois action is just the product over all$\ell$of the$\ell$-adic Tate modules$T_{\ell }(\operatorname{Jac}X)$. These are the$\ell$-adic analogues of the Hodge structures on first homomology, which bear the same information as the Jacobian itself. Thus it is not surprising (although very difficult!) that we can reconstruct$\operatorname{Jac}X$from these data, and the Jacobian determines the isomorphism class of the curve by Torelli's theorem. Now there are certainly some points where the above argument does not work as simply as presented, but the morals is that the analogue of the arithmetic fundamental group over$\bbc\$ should be the topological fundamental group with a "Hodge structure on groups". I do not know if this has ever been worked out, but there is a good understanding of the "Hodge structure on the nilpotent completion of the fundamental group", introduced by Hain and Zucker.