A visual explanation of why complex roots of polynomials occur in conjugate pairs:

The black parabola is the function

$z = (x-2)^2 + 1$

It clearly does not intersect the real x-axis, but has the pair of conjugate roots $$2 \pm i$$ , shown as red spheres. This can be understood by extending the real parabola over the complex plane using the pair of absolute value functions

$(x-2)^2 + (y \mp 1)^2$

where y is the imaginary part of the independent variable. Since the extension can happen in either direction, the solutions necessarily come in complex conjugate pairs.

Complete code for this example:

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