Relationship between algebraically closed fields and complete metric spaces?

The processes of algebraic closure and metric completion are actually less similar than they first appear. Metric completion is canonical in the sense that doing it requires making no arbitrary choices. You simply "adjoin limits of all Cauchy sequences." (More precisely - and this is not going to make sense to you, but I include it for the sake of completeness - there is an inclusion functor from the category of complete metric spaces to the category of metric spaces, and metric completion is its left adjoint.)

Algebraic closure, however, requires making certain arbitrary choices. It may not seem like it does, since you just "adjoin roots of all polynomials," but Cauchy sequences have unique limits and polynomials do not have unique roots; you (naively) have to pick some arbitrary order in which to adjoin the roots of a particular polynomial even though there's no canonical way to choose such an order. Moreover, you (naively) have to pick an order on the set of polynomials; you can't just adjoin all of their roots at once like you can with Cauchy sequences, since adjoining some roots will make adjoining the roots of other polynomials redundant. (More precisely - again, for the sake of completeness - there is an inclusion functor from the category of algebraically closed fields to the category of fields, and it does not have a left adjoint. Moreover, algebraic completion can't be extended to a functor at all.)

The algebraic closure of the rational numbers is an algebraically closed field, however it is only countable, so it is not $\mathbb C$.

However, there is still a Cauchy sequence converging to $\pi$ (since the rationals are dense in $\mathbb R$) but alas $\pi$ is not algebraic over $\mathbb Q$ so the field is not a complete metric space.

You may be interested in reading about the notions of real closed field and formally closed field which are somewhat related to your question.

On a general note, it seems that you see the similarity in "closure under property X", either Cauchy limits or polynomials. This is not a strange concept, and it is very common to take a certain property and ask yourself what happens when you close under it.

If you take the natural numbers, what happens when you close it under subtraction? You get $\mathbb Z$; when you close that under division you get $\mathbb Q$; and you can keep going and find richer structures (exponentiation, continuity, etc.)

The algebraic closure of $\mathbb{Q}$ is algebraically closed. Considering it as a subset of $\mathbb{C}$ it has an induced metric. It is not complete in this metric, for example because it contains $Q[i]$ which is dense in $\mathbb{C}$, but it is different from $\mathbb{C}$ (for example, because it is countable).