Here is a nice proof that was posted on the ALGTOP list recently. See also the ensuing discussion.
An algebraic extension of $\mathbb{C}$ is a unital division algebra over $\mathbb{C}$, say of dimension $n$, so induces $$\mathbb{CP}^{n-1} \times \mathbb{CP}^{n-1} \to \mathbb{CP}^{n-1}$$ satisfying $$ y \mapsto 1 \otimes y + y \otimes 1 $$ in second integral cohomology. Since $y^n = 0$, we must have $n=1$.
