Jacobian Matrix of complex integral

Question

Intuitively, If I define $z=\frac{x+iy}{\sqrt{2}}$ then the Jacobian matrix should be unity therefore:

$$\int\int dxdy=\int\int dz dz^*$$

however, I obtain the following result: $i$ using $$J=|\frac{\partial(z,z^*)}{\partial(x,y)}|$$