I just began to study about algebras over rings and quickly came across the fact that the quaternions are not an algebra over the complex numbers. I would prefer an answer as elementary as possible.
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.)