complex limits

edit

The limit is equivalent to two limits of functions of two real variables

x,y=var('x y',domain=RR)
z=x+I*y
f=(z.conjugate()/z)^2
f,  f.rectform()
((x - I*y)^2/(x + I*y)^2,
-4*I*(x^2 - y^2)*x*y/(x^2 + y^2)^2 - (4*x^2*y^2 - (x^2 - y^2)^2)/(x^2 + y^2)^2)

It does not exist, since the limits on some subsets are different

limit(f.subs(y=0),x=0),  limit(f.subs(x=y),y=0)
(1, -1)

Use

plot3d(real_part(f),(x,-1,1),(y,-1,1))
plot3d(imag_part(f),(x,-1,1),(y,-1,1))

for better understanding

Using another approach, if a is the modulus and b the argument, then

a,b=var('a b',domain=RR)
z=a*exp(I*b)
f=(z.conjugate()/z)^2;f

e^(-4*I*b)

and the limit

limit(f,a=0)
e^(-4*I*b)

depends on the argument , so the limit with z->0 does not exist