The eigenvalues and eigenvectors are going to be complex some of the time anyway. So I agree, why not just copy-and-paste the real algorithm (or use operator overloading) while remembering to insert complex-conjugates?
It's educational, sure. But it also seems unnecessary.
I think it's a very non-trivial task to "just" copy and paste a three-decade-old program that has tens of thousands of lines of dependencies, making sure to remember to add conjugates as necessary. :) Eigenvalues and eigenvectors are notoriously difficult programs to get right for general matrices with no known properties a priori.
It's educational, sure. But it also seems unnecessary.