Graphing Complex Solutions

The graph of f(x) = x^2 - 4x + 5 below in the xy-plane opens upward and has a vertex above the x-axis. Therefore, it has no real x-intercepts and the solutions to the equation when the function is set equal to zero turn out to be complex numbers. If you were to stretch out the x-axis into the complex number plane, you get the three dimensional graph shown here. Below the vertex, there is another parabola. It is in a plane that is perpendicular to the real x-axis. The two complex roots are the intercepts where this second parabola intersects the complex x-plane.