4. C.

According to the relationship between the roots and coefficients of a quadratic equation, we have
$$\left\{\begin{array}{l} p+q=-p, \\ p q=q . \end{array}\right.$$

When $q=0$, we have $p=q=0$, so $p q=0$;
When $q \neq 0$, we have $p=1, q=-2$, so $p q=-2$.