isomorphism between quotients of polynomial rings It is fairly known that 
$\mathbb C[x,y,z]/(xy+z^n) \cong \mathbb C[x,y,z]/(z^n+x^2+y^2)$. 
This appears, for example, in the study of singularities of type $A_n$. But, unfortunately, I am not able to prove it. 
How could we establish this isomorphism?
Thanks!

Hint:  The equality $x^2+y^2=(x+iy)(x-iy)$ suggests $x\mapsto x+iy$, $y\mapsto x-iy$, $z\mapsto z$.
