We consider a complex Hadamard matrix $H$ with entries in $\{\pm 1, \pm i\}$. I have seen some readings that often write $H = A + iB$ for matrices $A$ and $B$ with entries in $\{0,\pm 1\}$ where $A \circ B = 0$ ($\circ$ indicating the Hadamard product). I've never see this formulation of Hadamard matrices before, how might one prove that this is actually the case?
In particular, if $H$ is a complex Hadamard matrix, is it true to say that there are unique matrices $A$ and $B$ with entries in $\{0,\pm 1\}$ where $A \circ B = 0$ and $H = A+iB$? Even further, literature often uses this construction and says that the matrix $\begin{bmatrix}A+B & A-B \\ -A+B & A+B\end{bmatrix}$ is also a Hadamard matrix, how might we show that $A+B, A-B, -A+B$ have non zero entries to show this statement?