Hcfyk

In one of my previous answers, my first proposal

documentclass[tikz]standalone
usetikzlibraryquotes,angles
begindocument
begintikzpicture
draw (0,-4)--(0,4) node[above] $Im$ (-4,0)--(4,0) node[right] $Re$;
draw[dashed] (0,0) circle (3) circle (2);
coordinate (a) at (80:3);
coordinate (b) at (3,0);
coordinate (m) at (25:2);
coordinate (n) at (-95:2);
coordinate (p) at (145:2);
coordinate (o) at (0,0);
fill[black] (a) circle (2pt) (b) circle (2pt) (m) circle (2pt) (n) circle (2pt) (p) circle (2pt) (2,0) circle (2pt);
draw (a) node[above right] z;
draw (b) node[below] $;
draw (2,0) node[below] $;
draw (m) node[right] $;
draw (n) node[below] z;
draw (p) node[above] $;
draw (.1,1.5)--(0,1.5) node[left] $i$;
draw (1.5,.1)--(1.5,0) node[below] $1$;
draw (0,0)--(a) (0,0)--(m) (0,0)--(n) (0,0)--(p);
draw[dashed] (m)--(n)--(p)--cycle;
pic[draw,dashed,thick,"$theta$",angle radius=0.8cm,angle eccentricity=1.3] angle=b--o--a;
endtikzpicture
enddocument

gives this output

Seeing that it is a bit difficult to read some nodes (like the |z|^1/3e^{i (θ + 2 π)/3} one), I redefine the inner sep and set the fill color of the nodes. Eventually I get

documentclass[tikz]standalone
usetikzlibraryquotes,angles,positioning
begindocument
begintikzpicture
beginscope[every node/.style=fill=white,inner sep=2pt]
draw (0,-4)--(0,4) node[above] $Im$ (-4,0)--(4,0) node[right] $Re$;
draw[dashed] (0,0) circle (3) circle (2);
coordinate (a) at (80:3);
coordinate (b) at (3,0);
coordinate (m) at (80/3:2);
coordinate (n) at (80/3-120:2);
coordinate (p) at (80/3+120:2);
coordinate (o) at (0,0);
draw (a) node[above right] z;
draw (b) node[below right] $;
draw (2,0) node[below left=0cm and -2em] $;
draw (m) node[right] $;
draw (n) node[below] z;
draw (p) node[above] $;
draw (.1,1.5)--(0,1.5) node[left] $i$;
draw (1.5,.1)--(1.5,0) node[below] $1$;
draw (0,0)--(a) (0,0)--(m) (0,0)--(n) (0,0)--(p);
draw[dashed] (m)--(n)--(p)--cycle;
endscope
pic[draw,dashed,thick,"$theta$",angle radius=0.8cm,angle eccentricity=1.3] angle=b--o--a;
fill[black] (a) circle (2pt) (b) circle (2pt) (m) circle (2pt) (n) circle (2pt) (p) circle (2pt) (2,0) circle (2pt);
endtikzpicture
enddocument

The separation of the nodes and the paths are now quite good IMHO, except some cases where the separation is over-made:

A solution to it is to change the default rectangle node shape

to something like this

^{Sorry, I am not good at drawing, especially drawing with a computer mouse.}

In other word, I think I have to create a new TikZ node shape which depends on the maximum and the minimum "y-coordinates" of all characters.

It is way too complicated to me, and I haven't found a hint on this.

Can you help me? Any help is much appreciated.

Maybe the contour package is a solution for you (See also this answer)?

documentclass[tikz, border=2mm]standalone

usepackage[outline]contour
contourlength2pt % increase the white space

usetikzlibrarypatterns

begindocument

begintikzpicture
draw [step=.25] (-2,-.5) grid (2,.5);

node[] at (0,0) contourwhite$;

endtikzpicture

enddocument

Note: The above does not work with XeLaTeX or LuaLaTeX. In this case you need to remove the outline option and add for example contournumber60 (or a higher number, depending on the desired smoothness of the edges of the white space, see the documentation of the package).