Open quantum systems are good models of many interesting physical systems. Non-Hermitian Hamiltonians are known to describe, or at least approximate some of these open quantum systems well. Recently, there has been an increase in interest in quantum algorithms for simulating such Hamiltonians, such as the Quantum Imaginary Time Evolution algorithm, and other ones based on trace preserving quantum operations, using an enlarged Hilbert space. The focus of our work is on testing the near-term applicability of some of these NISQ-era algorithms on real, noisy quantum hardware. We will look at the 1D quantum Ising model in complex parameter space. Such models have a rich phase-structure in the complex plane and studying them would allow us to explore critical regions such as Lee-Yang edges and Fisher zeros. We will also discuss the applicability of these algorithms for ground-state preparation.