6. Let the complex number $z$ satisfy
$$
|z-| z+1||=|z+| z-1|| \text {. }
$$

Then the incorrect statement among the following is ( ).
(A) $z$ could be a pure imaginary number
(B) $z$ could be a real number
(C) The real part of $z$ is less than 2
(D) The argument of $z$ could be $\frac{\pi}{4}$
6. D.

Squaring both sides of the original equation, we get
$$
\begin{array}{l}
(z-|z+1|)(\bar{z}-|z+1|) \\
\quad=(z+|z-1|)(\bar{z}+|z-1|) .
\end{array}
$$

Notice that, $|z+1|^{2}-|z-1|^{2}=2(z+\bar{z})$.
Expanding and rearranging equation (1), we get
$$
(z+\bar{z})(|z+1|+|z-1|-2)=0 \text {. }
$$

From this, either $z+\bar{z}=0$, meaning $z$ is a pure imaginary number; or $|z+1|+|z-1|=2$, meaning $z$ is a real number, and lies between -1 and 1.