\newcommand\markangle[6]{% origin X Y radius radiusmark mark
  % fill red circle
  \begin{scope}
    \path[clip] (#1) -- (#2) -- (#3);
    \fill[color=red,fill opacity=0.5,draw=red,name path=circle]
    (#1) circle (#4);
  \end{scope}
  % middle calculation
  \path[name path=line one] (#1) -- (#2);
  \path[name path=line two] (#1) -- (#3);
  \path[%
  name intersections={of=line one and circle, by={inter one}},
  name intersections={of=line two and circle, by={inter two}}
  ] (inter one) -- (inter two) coordinate[pos=.5] (middle);
  % put mark
  \node at ($(#1)!#5!(middle)$) {#6};
}

\begin{tikzpicture}
    \newcommand{\R}{2}
    \draw (0,0) circle (\R);
    \draw[->, thick] ({-(\R+0.2)},0) -- ({\R+0.2},0);
    \draw[->, thick] (0,{-(\R+0.2)}) -- (0,{\R+0.2});
    \draw[thick] (\R,-0.06) -- (\R,0.06) node[label=below:$1$] {};
    \draw[thick] (-0.06,\R) -- (0.06,\R) node[label=left:$i$] {};
    \draw (0,0) -- ({\R*cos(30)},{\R*sin(30)}) node[label=15:$z$] {};
    \draw (0,0) -- ({\R*cos(60)},{\R*sin(60)}) node[label=35:$z^2$] {};

    \coordinate (O) at (0,0);
    \coordinate (X) at (1,0);
    \coordinate (Y) at ({\R*cos(30)},{\R*sin(30)});
    \coordinate (Z) at ({\R*cos(60)},{\R*sin(60)});
    \markangle{O}{Y}{Z}{10mm}{7mm}{$\varphi$}
    \markangle{O}{X}{Y}{10mm}{7mm}{$\varphi$}

    \begin{scope}[xshift=4cm, yshift=-1.2cm]
        \draw (0,0) circle (\R/2);
        \newcommand{\x}{1}
        \draw [red,thick,domain=-30:30] plot ({cos(\x)}, {sin(\x)});
        \draw [red,thick,domain=210:150] plot ({cos(\x)}, {sin(\x)});
    \end{scope}

    \begin{scope}[xshift=4cm, yshift=+1.2cm]
        \draw (0,0) circle (\R/2);
        \newcommand{\x}{1}
        \draw [red,thick,domain=-30:30] plot ({cos(\x)}, {sin(\x)});
    \end{scope}

    \coordinate (T) at (5.5,1);
    \path[->] (5.5,-1) edge[bend right] node[label=right:$z^2$] {} (T);
\end{tikzpicture}