Brahmagupta-Fibonacci Identity/Extension/Proof 3

Proof
Let $z_j = a_j + i b_j$ for each $j = 1, 2, \ldots, n$.

Let $\ds c + i d = \prod_{j \mathop = 1}^n z_j$.

Then:

As $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are integers, so are $c$ and $d$.

Thus: