The function he showed _is_ an isomorphism, but those 4 properties he showed above don't have anything to do with that (and they are also not used to prove that that function is an iso).
EDIT: Very stupid mistake, it is not a isomorphism because it is not onto. For example the matrix (a)ij = 1 for i=j={1,2} can not be obtained.
Of course, his function isn't even close to bijective (it's not onto), so it's just a homomorphism, not an isomorphism.
I mean, other than mathematicians, no one cares what an isomorphism is, right? But doing complex arithmetic with matrices sounds cool because one can code stuff like Fourier transforms without a library that implements complex numbers. I found that refreshing and interesting and that is why I submitted this article.
Most languages make it pretty easy to implement basic complex number support.