The equation \(x^2 + y^2 = 1\) describes the unit circle in the real Cartesia
n plane. We may write \(y = \pm\sqrt{1 - x^2}\) and treat the circle as a union
of two semi-circle graphs.
Similarly, the equation \(z^2 + w^2 = 1\) in the complex Cartesian plane describes a surface that can be written as a union of two graphs. The complex square root is only continuous if we remove branch cuts, and the two graphs here are only continuous if we slit the complex plane somewhere.
This image shows continuous branches on the plane with the real interval \([-1, 1]\) removed.
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