$\mathbb{C}$ is closure of residue field modulo infintely large prime

It is a well-known fact that there exists a non‑principal ultrafilter $\mathcal{U}$ on the set of prime numbers such that the ultraproduct of the algebraic closures of the finite fields is isomorphic to the complex numbers: $$ \mathbb{C}\;\cong\; \prod_{\mathcal{U}}\ \overline{\mathbb{F}}_p . $$ The standard proof relies on the observation that both are algebraically closed fields of characteristic $0$, of cardinality $2^{\aleph_0}$, and have infinite transcendence degree over $\mathbb{Q}$.

This isomorphism has greatly clarified many reduction‑mod‑$p$ arguments for me. I often picture it geometrically in the following way: the “one‑dimensional” ultraproduct $\prod_{\mathcal{U}} \mathbb{F}_p$ (before taking algebraic closures) corresponds to the real line $\mathbb{R}$, and then adjoining algebraic elements (roots of unity, etc.) to obtain the full ultraproduct $\prod_{\mathcal{U}}\overline{\mathbb{F}}_p$ corresponds to passing from $\mathbb{R}$ to $\mathbb{C}$. In this mental picture, roots of unity in the ultraproduct are mapped to roots of unity in $\mathbb{C}$.

I have several questions about this viewpoint:

1. Validity of the analogy. Can this geometric picture be made rigorous? For instance, does there exist a subfield of $\prod_{\mathcal{U}}\overline{\mathbb{F}}_p$ that is isomorphic to $\mathbb{R}$ and that plays the role of the “real line” in the ultraproduct? More generally, is there a way to formalise the idea that $\prod_{\mathcal{U}}\mathbb{F}_p$ sits inside $\prod_{\mathcal{U}}\overline{\mathbb{F}}_p \cong \mathbb{C}$ like $\mathbb{R}$ sits inside $\mathbb{C}$?

2. Other visual interpretations. Are there alternative conceptual or visual ways to understand this isomorphism? Any different mental images that illuminate how properties transfer between characteristic $p$ and characteristic $0$ would be very welcome.

3. Visualising the Frobenius. Under this isomorphism, can one “see” the Frobenius endomorphism, and consequently the Frobenius twist? That is, how does the family of Frobenius maps $x \mapsto x^p$ on each $\overline{\mathbb{F}}_p$ behave after taking the ultraproduct and identifying the result with $\mathbb{C}$? Does the resulting automorphism of $\mathbb{C}$ admit a geometric interpretation?

I am a graduate student, so feel free to use arguments of any level of technicality.