6. Let the complex number $z$ satisfy $|z-| z+1||=|z+| z-1||$, then the incorrect judgment 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 $\frac{\pi}{4}$
Let $z=x+y \mathrm{i}(x, y \in \mathbf{R})$, then $\left|x-\sqrt{(x+1)^{2}+y^{2}}+y \mathrm{i}\right|=\left|x+\sqrt{(x-1)^{2}+y^{2}}+y \mathrm{i}\right|$ $\Rightarrow\left(x-\sqrt{(x+1)^{2}+y^{2}}\right)^{2}=\left(x+\sqrt{(x-1)^{2}+y^{2}}\right)^{2}$
$\Rightarrow x\left(\sqrt{(x+1)^{2}+y^{2}}+\sqrt{(x-1)^{2}+y^{2}}-2\right)=0$, thus the point $Z$ corresponding to the complex number $z$ is on the $y$-axis or on the line segment connecting $(-1,0),(1,0)$, hence $A, B, C$ are correct, and $D$ is incorrect. Therefore, the answer is $D$.