Complex function grid transformation

I have a grid and a complex function of one variable :

x=linspace(0,1,10);
y=linspace(0,1,10);
f=@(z) z.^2; %for example f(1+2i)=-3+4i

I want to plot the images under f of the constant curves for and also the constant curves for , but without using plot. This is what I did in my code:

hold on
for i=1:10
        plot(real(f(x(i)+1i.*y)),imag(f(x(i)+1i.*y)));
end
for j=1:10
     plot(real(f(x+1i.*y(j))),imag(f(x+1i.*y(j))));
end
hold off

and I get the following result:

Is it posible to better represent the image of this square under a complex function? Are there some more specific commands? How can we define a function as a complex function?