As spatial division multiplexed transmission systems employing few mode fibers (FMFs) rely heavily on digital signal processing (DSP), the impact of computational complexity should be considered. A key DSP process block is the carrier phase estimation (CPE), which consists of a phase estimator and digital phase locked loop (DPLL) per output. In this work, a low complexity phase estimator is proposed, and the common-mode laser frequency offset is exploited to reduce the number of DPLLs. The combination of a low complexity CPE and single DPLL is experimentally demonstrated for a 28GBaud 6-channel transmission over 41.7km FMF, carrying up to 32 quadrature amplitude modulation (QAM). Thereby, the nonlinear tolerances for the proposed CPE scheme is shown to perform similarly as the more computationally complex conventional CPE scheme.