The complex number z satisfying ${\mathrm{z}}^2 + \left|\mathrm{z}\right| = 0$ is

$$\begin{gathered} {\mathrm{z}}^2+\left|\mathrm{z}\right|=0 \\ \mathrm{z}=\mathrm{x}+\mathrm{iy}\Rightarrow {(\mathrm{x}+\mathrm{iy})}^2+|x+\mathrm{iy}|=0 \\ \therefore {\mathrm{x}}^2-{\mathrm{y}}^2+\mathrm{i}2\mathrm{xy}+\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=0 \\ \Rightarrow \left({\mathrm{x}}^2-{\mathrm{y}}^2+\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}\right)+\mathrm{i}\left(2\mathrm{xy}\right)=0 \\ \Rightarrow {\mathrm{x}}^2-{\mathrm{y}}^2+\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=0, \mathrm{xy}=0 \\ \boxed{\mathrm{x}=0}\Rightarrow -{\mathrm{y}}^2+\left|\mathrm{y}\right|=0 \Rightarrow {\mathrm{y}}^2\pm \mathrm{y}=0 \\ \Rightarrow \mathrm{y}\left(1+y\right)=0, \mathrm{y}\left(1-y\right)=0 \\ \therefore \mathrm{y}=0, \mathrm{y}=1, \mathrm{y}=-1 \\ \Rightarrow \mathrm{z}=\left(0,0\right),\left(0,1\right),\left(0,-1\right) \\ \boxed{\mathrm{z}=0, \pm \mathrm{i}} \end{gathered}$$