Based on the existing answers, which opened my eyes for what's going on here, I would like to present yet another very simple expression for the solution, which is just slightly different from the one in AlexTP's answer (and which turned out to be equivalent to the one given in Jason R's answer, as shown below in the EDIT-part).
[EDIT: now that AlexTP has edited his answer, our expressions for the PDF are identical; so all three answers finally agree with each other].
Let the complex random variable Z=X+jY be defined as
Z=ρejθ(1)
where the radius ρ is deterministic and given, whereas the angle θ is random and uniformly distributed on [0,2π). I state without further proof that Z is circularly symmetrical, from which it follows that its probability density function (PDF) must satisfy
fZ(z)=fZ(x+jy)=fZ(r),withr=x2+y2−−−−−−√(2)
i.e., it can be written as a function of the radius (magnitude) r.
Since the PDF must be zero everywhere except for r=ρ, and since it must integrate to unity (when integrated over the 2-dimensional plane), the only possible PDF is
fZ(r)=12πδ(r−ρ)(3)
It can be shown that (3) leads to the correct marginal densities for the random variables X and Y.
EDIT:
After some very useful discussion in the comments it appears that we've managed to agree on one solution to the problem. I will show in the following that the unassuming formula (3) is actually equivalent to the more involved looking formula in Jason R's answer. Note that I use r for the magnitude (radius) of the complex RV Z, whereas in Jason's answer r denotes the real part of Z. I will use x and y for the real and imaginary parts, respectively. Here we go:
fZ(r)=12πδ(r−ρ)=12πδ(x2+y2−−−−−−√−ρ)(4)
We know that δ(g(x)) is given by
δ(g(x))=∑iδ(x−xi)|g′(xi)|(5)
where xi are the (simple) roots of g(x). We have
g(x)=x2+y2−−−−−−√−ρandg′(x)=xx2+y2−−−−−−√=xr(6)
The two roots xi are
x1,2=±ρ2−y2−−−−−−√(7)
Consequently,
|g′(x1)|=|g′(x2)|=ρ2−y2−−−−−−√ρ=1−(yρ)2−−−−−−−−√(8)
With (5)-(8), Eq. (4) can be written as
fX,Y(x,y)=12π1−(yρ)2−−−−−−−−√[δ(x−ρ2−y2−−−−−−√)+δ(x+ρ2−y2−−−−−−√)](9)
For ρ=1, Eq. (9) is identical to the expression given in Jason R's answer.
I think we can now agree that Eq. (3) is a correct (and very simple) expression for the PDF of the complex RV Z=ρejθ with deterministic ρ and uniformly distributed θ.