Here is another short proof, showing that no finite extension of $\mathbb{C}$ exists. If $A$ were such an extension of dimension $d$, then the projective space $\mathbb{P] \mathbb{P} A \cong \mathbb{CP}^{d-1}$ were a compact commutative Lie group, hence a torus. There are plenty of ways showing that $\mathbb{CP}^{m}$ is not a torus (if $m>0$). Take your favorite one, and the proof is complete.
Here is another short proof, showing that no finite extension of $\mathbb{C}$ exists. If $A$ were such an extension of dimension $d$, then the projective space $\mathbb{P] A \cong \mathbb{CP}^{d-1}$ were a compact commutative Lie group, hence a torus. There are plenty of ways showing that $\mathbb{CP}^{m}$ is not a torus (if $m>0$). Take your favorite one, and the proof is complete.