The equation \(x^2 + y^2 = 1\) describes the unit circle in the real Cartesian 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, one green, the other blue, on the plane with the real rays \((-\infty, -1]\) and \([1, \infty)\) removed.
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