In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ζ = ∑ k =0,...,∞ 23 - k ε η k ≡ Δ η 1 ,...$,η k where the indices η k are independent random variables taking the values 0,1,2,3,4 with probabilities p 0 k , p 1 k , p 2 k , p 3 k , p 4 k respectively, p ik ≥ 0, p ik ≥ 0, ∑ i =0,...,4 p ik = 1 for any k ∈ ℕ, ε m = e m π i /2 = i m ( m = 0,1,2,3) are 4th roots of unity, ε 4 = 0. We prove that the distribution of ζ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of ζ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found.