When changing variables in a real integral, we use the absolute value of the Jacobian determinant to guarantee that orientations are preserved in the new set of coordinates. The determinant is a continuous function and, if it starts positive and is non-vanishing, it cannot reach negative values and change orientations. Thus, if we change coordinates in an integral and the Jacobian determinant starts positive and is never $0$, the absolute value is unnecessary, since orientations will be always preserved.
I want to generalize this to complex manifolds. I know that holomorphic maps preserve orientation, but I'm dealing with a non-holomorphic map in my work. Its Jacobian determinant has the form $\sqrt{x + iy}$, starts at $1$ and moves around quite a bit (it changes branch several times), but it is always non-vanishing. Will it preserve orientation?