Polynomial Roots Under Conjugate Map

Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\bar{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\bar{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1, z_2, z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ 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 $B + D$? (A) $-304$ (B) $-208$ (C) $12i$ (D) $208$ (E) $304$