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: