Why are the quaternions not an algebra over the complex numbers?

If it were a $\Bbb C$ algebra, it would have to be dimension $2$ and contain a copy of $\Bbb C$.

Taking any $x$ not in the copy of $\Bbb C$ situated in $\Bbb H$, the span of $\Bbb C$ and $x$ is the whole ring. But products between elements of this span commute with each other, and that means the span is a commutative ring. This contradicts the fact the quaternions aren't commutative.

At another level, the Artin-Wedderburn theorem says that the only possible simple Artinian $\Bbb C$-algebras are the square matrix rings over $\Bbb C$, but none of them have dimension $2$. ($\Bbb H$ is a simple Artinian ring.)