Complex structure of a real Clifford algebra

The classification of real Clifford algebras in terms of matrix algebras is well–known. Here we consider the real Clifford algebra $\mathcal{C}l(r,s)$ not as a matrix algebra, but as a Clifford module over itself. We show that $\mathcal{C}l(r,s)$ possesses a basis independent complex structure only when the square of the volume element $\omega$ is -1, in which case it is uniquely given up to sign by right multiplication with $\omega$.