4. The number of distinct roots of the equation $a x^{2}+b \mid x+c=0(a, b, c \in \mathbf{R}, a \neq 0)$ in the set of complex numbers is ().
A. 2 or 4
B. $1,2,3,4$
C. At most 6
D. Possibly 8
4. C
Let $x=m+n i(m, n \in \mathbf{R})$, substitute into the equation, we get
$$
\begin{array}{l}
\left\{\begin{array}{l}
2 m n=0, \\
a\left(m^{2}-n^{2}\right)+b \sqrt{m^{2}+n^{2}}+c=0,
\end{array}\right. \\
\Leftrightarrow\left\{\begin{array} { l } 
{ m = 0 , } \\
{ - a n ^ { 2 } + b | n | + c = 0 ; }
\end{array} \text { or } \left\{\begin{array}{l}
n=0, \\
a m^{2}+b|m|+c=0 .
\end{array}\right.\right.
\end{array}
$$

In the equations $a m^{2}+b|m|+c=0$ and $-a n^{2}+b|n|+c=0$, if one equation has 4 real roots, then the other has at most 2 real roots (this can be seen from the corresponding function graphs).