I've previously asked elsewhere, https://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a programming language. I am still working with a 2x2 matrix ($A$) but it's now complex, and takes the form: $$ A = \begin{bmatrix} a+jb & c + jd \\ e + jf & g +jh \\ \end{bmatrix} $$ where $j$ is the square root of $-1$.
Do the equations in http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html (which is used as a part of the solution in the previous link) still hold?
Yes, the formula does still work. It was found by solving the quadratic equation, and polynomials with complex coefficients can be factored in this way.