Complex branches of a generalised Lambert $W$ function arising from $p,q$--binomial coefficients

The $\psi (x)$-function, which solves the equation $x=\sinh (aw)e^w$ for $0<a<1$, has a natural connection to the renowned Lambert $W$ function and also physical relevance through its connection to the Lenz-Ising model of ferromagnetism. We give a detailed analysis of its complex branches and construct Riemann surfaces from these under various conditions of $a$, unveiling intriguing new links to the Lambert $W$ function.