Is there a connection between the analytical properties of complex analytic functions and the complex numbers being complete algebraically?

Question

I will first have to apologize that this will be a very fuzzy question. At this time I have no better way to formulate it. I am willing to reformulate it when I can better pinpoint what it is that gives me this idea.

Is there a connection between the analytic properties of complex valued functions and the complex numbers being complete algebraically?

Meromorphic functions we can write as

$$f(z) = \frac{\prod_k (z-h_k)}{\prod_k (z-p_k)}$$

We can show that any expression of the form (thanks to the fund theorem of algebra)

$$\frac{1}{f(z)}+\frac{1}{g(z)}=\frac{h(z)}{f(z)g(z)}$$

Where $z\to h(z)$ is ensured to be factorable.

Does this have any connection to the overly nice properties of complex analytic functions?

Answer 1

One of my teachers always attributed the nice behavior of analytic functions to two things:

1. The fact that the invertible elements are a connected set (so that you can always take paths around the one non-invertible element)
2. Complex differentiability is "much stronger" than real differentiability (because you are able to "approach from all directions in 2-d" as opposed to just from two sides in 1-d)

I do not know if either of these two things are directly related to algebraic completeness or if it is just a result that $\mathbb C$ is the only extension to be had for $\mathbb R$.