1. Given the function defined on the set of complex numbers $f(z)=(4+\mathrm{i}) z^{2}+p z+q(p, q$ are complex numbers $)$. If $f(1)$ and $f(\mathrm{i})$ are both real numbers, then the minimum value of $|p|+|q|$ is $\qquad$ .
$-1 . \sqrt{2}$.
$$
\begin{array}{l}
\text { Let } p=a+b i, q=c+d i(a, b, c, d \in \mathbf{R}) . \\
\text { From } f(1)=(4+a+c)+(1+b+d) i, \\
f(i)=(-4-b+c)+(-1+a+d) i
\end{array}
$$

being real, we know $a=1-d, b=-1-d$.
$$
\begin{array}{l}
\text { Then }|p|+|q|=\sqrt{a^{2}+b^{2}}+\sqrt{c^{2}+d^{2}} \\
=\sqrt{2+2 d^{2}}+\sqrt{c^{2}+d^{2}} \text {. }
\end{array}
$$

Therefore, when $c=d=0$ (i.e., $a=1, b=-1$), $|p|+|q|$ achieves its minimum value $\sqrt{2}$.