A convergence test of Pringsheim type for the branching continued fraction (*) \(^{\infty}_{k=1}\sum^{N}_{i(k)=1}\frac{a(i(k))}{b(i(k))}\) with complex number elements is derived. It is shown that if \(| b(i(k))| \geq N | a(i(k))| +1\) for all corresponding partial denominators and numerators b(i(k)) and a(i(k)) occurring in (*), then this expansion converges to z, where \(| z| \leq 1\). Special conditions ensuring that \(| z| =1\) are stated.