In the book Spin Geometry of Lawson was mentioned that a complex representation of $Cl_{r,s}$ is a real representation $\rho:Cl_{r,s}\to\mathrm{Hom}_\mathbb{R}(W,W)$ such that
$\rho\circ J(\phi)=J\circ\rho(\phi)$ for all $\phi\in Cl_{r,s}$.
Note that $J$ is a real linear map from $W$ to itself s.t. $J^2=-\mathrm{Id}$. I know that I can make a real vector space $W$ with this map to a complex vector space. My idea was to show that the above relation implies $\rho(\phi)((a+ib)w)=(a+ib)\rho(\phi)(w)$.
Here is one calculation I was trying $(\rho(\phi)\circ J)((a+ib)w)=(\rho(\phi)\circ J)((aw+bJ(w))=\rho(aJ(w)-bw)=a\rho(J(w))-b\rho(w)\overset{!}{=}(J\circ\rho(\phi))((a+ib)w)$.
Thanks for your help.