Let $A,B$ real $n \times n$ matrices such that $A+iB$ is invertible.

I want to prove (if true) that $|A+iB|\cdot|A-iB| >0$

Of course, this is the same as $\big| (A+iB)(A-iB) \big|>0$, so, multiplying them gives $C= A^2 + iBA -iAB + B^2$

Can we say anthing about the determinant of $C$?