It is normal...(you should remember that a matrix is normal if and only if it is unitarily diagonalisable).

It is normal as if

(1 2i)

(2 1)=A

and

(1 2)

(-2i 1)=B

then A and B commute. As B is the conjugate transpose thing of A, you are done...

(I've left the LaTeX in in the hope that the compiler will start working again...)