A question on the construction of the complex numbers and field extensions I am thinking about a construction of the complex numbers. I know that it is not necessary: for a complex analysis course one could just give the field axioms and then take $\mathbb C$ to be a field that satisfies them. 
But since we can construct the real numbers as set of equivalence classes of Cauchy sequences in $\mathbb Q$ I started to think about how to do a construction for $\mathbb C$.
I found that $\mathbb C$ can be constructed as the field extension  $\mathbb R[x]/\langle x^2 + 1\rangle$ of $\mathbb R$. 
My question is: 

Does this construction give a unique extension field? (are extension
  fields unique? I could not find any information in the affirmative)
And if not, how can one prove after constructing $\mathbb C$ like this
  that $\mathbb C$ is the unique field with the stipulated properties?


*

*Up to isomorphism, $\mathbb C$ is the only finite extension of $\mathbb R$. This follows from the fact that $\mathbb C$ is algebraically closed, that is, the fundamental theorem of algebra.

*There are many others extensions of $\mathbb R$, such as $\mathbb R(X)$, the field of rational functions with real coefficients.
