Recall that the conjugate of the complex number $w=a+b i$, where $a$ and $b$ are real numbers and $i=\sqrt{-1}$, is the complex number $\bar{w}=a-b i$.
For any complex number $z$, let $f(z)=4 i \bar{z}$. The polynomial
$P(z)=z^{4}+4 z^{3}+3 z^{2}+2 z+1$
has four complex roots: $z{1}, z{2} z{3}$, and $z{4}$. Let
$Q(z)=z^{4}+A z^{3}+B z^{2}+C z+D$
be the polynomial whose roots are $f(z{1}), f(z{2}), f(z{3})$, and $f(z{4})$, where the coefficients $A, B, C$, and $D$ are complex numbers. What is $B+D ?$