4. If the complex coefficient equation with respect to $x$
$$
(1+2 \mathrm{i}) x^{2}+m x+1-2 \mathrm{i}=0
$$

has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$.
4. 2 .

Let $\alpha$ be a real root of the original equation, $m=p+q$ i.
$$
\left.\begin{array}{l}
\text { Then }\left(1+2 \text { i) } \alpha^{2}+(p+q \text { i) } \alpha+1-2 \mathrm{i}=0\right. \\
\Rightarrow\left\{\begin{array}{l}
\alpha^{2}+p \alpha+1=0, \\
2 \alpha^{2}+q \alpha-2=0
\end{array}\right. \\
\Rightarrow\left\{\begin{array}{l}
p=-\left(\alpha+\frac{1}{\alpha}\right), \\
q=-2\left(\alpha-\frac{1}{\alpha}\right)
\end{array}\right. \\
\Rightarrow|m|^{2}=p^{2}+q^{2}
\end{array}\right\}
$$

When and only when $\alpha^{4}=1$, i.e., $\alpha= \pm 1$, the equality holds, at this time, $m= \pm 2$.
Therefore, the minimum value of $|m|$ is 2.