$f\in C[[x+iy]]$ means $f(x,y)=g(x+iy),$ where $g\in C[[z]]$. Then
$$f_x=g'(x+iy),\; f_y=ig'(x+iy),$$
therefore $f_y=if_x$ and this is Cauchy-Riemann: indeed, if $f=u+iv$ then
$f_x=u_x+iv_x$ and $f_y=u_y+iv_y$, so $f_y=if_x$ is equivalent to  
$u_x=v_y$, $u_y=-v_x$.

To prove the converse, make the linear change of the variables:
$z=x+iy,\; \overline{z}=x-iy$. Then the Cauchy-Riemann conditions in new variables become $f_{\overline{z}}=0$.
Assuming that this is satisfied, set $g=f_z$. Then $$g_{\overline{z}}=f_{z\overline{z}}=f_{\overline{z}z}=0,$$
so $g$ does not depend on $\overline{z}$ that is $g\in C[[z]]$.

Remark. This is how old textbooks like Whittaker Watson explain the Cauchy-Riemann conditions. Pure algebra.