4. Given a complex number $z \neq 0$. Provide the following propositions:
(1) $z+\bar{z}=0$;
(2) $z^{2}=-a^{2}(a \in \mathbf{R}, a \neq 0)$;
(3) $z^{2}=z^{2}$;
(4) $|z+a|=|z-a|(a \in \mathbf{R}, a \neq 0)$.
Among the above propositions, which one is a sufficient and necessary condition for $z$ to be a pure imaginary number?
4.(1)、(2)、(4).
$z+\bar{z}=0 \Leftrightarrow \operatorname{Re}(z)=0 \Leftrightarrow z$ is a pure imaginary number.
$=-a i \Leftrightarrow z$ will be purely imaginary.
$$
\begin{array}{c}
|z+a|=|z-a| \Leftrightarrow \sqrt{(x+a)^{2}+y^{2}}= \\
\sqrt{(x-a)^{2}+y^{2}} \Leftrightarrow(x+a)^{2}=(x-a)^{2} \Leftrightarrow 2 a x=0 \Leftrightarrow
\end{array}
$$
$x=0$ and $z$ is a pure imaginary number.
$$
z^{2}-\bar{z}^{2}=0 \Leftrightarrow(z-\bar{z})(z+\bar{z})=0 \Leftrightarrow z=\bar{z} \text { or } z+
$$
$:=0 \Leftrightarrow z$ is a pure imaginary number.