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.)