Exercises with Complex Functions

z[n_]:=Exp[I*Pi*n*Log[n/8]*2^0.5];
L={{0,0}}; n1=100;n2=1500;
Do[L=Append[L,{Re[L[[i,1]]+I*L[[i,2]]+z[i]],
Im[L[[i,1]]+I*L[[i,2]]+z[i]]}],{i,n2}];
LL=Table[{Hue[(i-n1)/n2],
Line[{{k1*L[[i,1]],k2*L[[i,2]]},
{k1*L[[i+1,1]],k2*L[[i+1,2]]}}],
Hue[1-(i-n1)/n2],
Line[{{k1*L[[i,2]],k2*L[[i,1]]},
{k1*L[[i+1,2]],k2*L[[i+1,1]]}}]},
{i,n1,n2},{k1,{-1,1}},{k2,{-1,1}}];
Graphics[LL,ImageSize->500,
Background->RGBColor["slategray"]]