The paper establishes new convergence domains for branched continued fraction expansions of Horn's hypergeometric function $ H_4 $ with real and complex parameters. These domains enable the PC method to extend analytical functions to their expansions in the studied convergence regions. The function $ H_4 $ is defined as a double series with specific conditions on parameters. The paper extends previous works by analyzing the convergence of branched continued fractions for specific ratios of $ H_4 $, proving their convergence in certain domains. Theorems 2, 3, and 4 provide convergence criteria for different types of $ h_k $ values. The PC method is used to establish analytical continuation of the function in these domains. Examples illustrate the convergence and analytic continuation of $ H_4 $ in specific regions. The study also discusses the application of branched continued fractions in various areas of mathematics and physics. The results contribute to the understanding of the analytic continuation of special functions through branched continued fractions.The paper establishes new convergence domains for branched continued fraction expansions of Horn's hypergeometric function $ H_4 $ with real and complex parameters. These domains enable the PC method to extend analytical functions to their expansions in the studied convergence regions. The function $ H_4 $ is defined as a double series with specific conditions on parameters. The paper extends previous works by analyzing the convergence of branched continued fractions for specific ratios of $ H_4 $, proving their convergence in certain domains. Theorems 2, 3, and 4 provide convergence criteria for different types of $ h_k $ values. The PC method is used to establish analytical continuation of the function in these domains. Examples illustrate the convergence and analytic continuation of $ H_4 $ in specific regions. The study also discusses the application of branched continued fractions in various areas of mathematics and physics. The results contribute to the understanding of the analytic continuation of special functions through branched continued fractions.