  We study the correlations of pairs of complex logarithms of $\mathbb
Z$-lattice points in the complex line at various scalings, proving the
existence of pair correlation functions. We prove that at the linear scaling,
the pair correlations exhibit level repulsion, as it sometimes occurs in
statistical physics. We prove total loss of mass phenomena at superlinear
scalings, and Poissonian behaviour at sublinear scalings. The case of Euler
weights has applications to the pair correlation of the lengths of common
perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to
itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb
H^3_{\mathbb R}$.
