The paper considers the extension of analytic functions by a special family of functions — branched continued fractions. The truncation error bounds for branched continue fraction expansions of the Horn's hypergeometric functions $H_4$ and their ratios with certain conditions on real parameters are established. In this case, a new domain of analytical continuation of these functions is also established using the PF method (based on the so-called property of fork for approximants of a branched continued fraction).