\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
    xmin=-3,xmax=3,
    ymin=-3,ymax=3,
    axis lines=center,
    axis line style={latex-latex},
    axis equal,
    grid=both,
    xlabel={$\operatorname{Re}(z)$},
    ylabel={$\operatorname{Im}(z)$},
    xlabel style={right},
    ylabel style={above},
    ticks=both,
    enlargelimits={abs=0.5},
    clip=false,
]
\addplot[color=black,mark=*] coordinates {(1,2)};
\draw[->,black,thick] (0,0) -- (1,2);
\addplot[color=black,mark=*] coordinates {(-1, 2)};
\draw[->,black,thick] (0,0) -- (-1,2);
\addplot[color=black,mark=*] coordinates {(-1,-2)};
\draw[->,black,thick] (0,0) -- (-1,-2);
\addplot[color=black,mark=*] coordinates {(1,-2)};
\draw[->,black,thick] (0,0) -- (1,-2);
\node[above right] at (1,2) {$z_1$};
\node[above left] at (-1,2) {$z_2$};
\node[below left] at (-1,-2) {$z_3$};
\node[below right] at (1,-2) {$z_4$};
\end{axis}
\end{tikzpicture}
\end{document}