Thermodynamics of the quark-gluon plasma at finite density is studied in the framework of the Field Correlator Method, where thermodynamical effects of Polyakov loops and the colormagnetic confinement force (CMC) is defined by the spatial Wilson loop projection are taken into account. It was shown that the CMC potential plays an important role in QGP thermodynamics providing the magnetic Debye mass for quarks and gluons. Having found good agreement with numerical lattice data for zero density, an analytical expression for the pressure P (T , μ) was calculated for 0 < μ < 400 MeV and 150 < T < 1000 MeV. The analytical structure of the expression for the QGP equation of state was studied in the complex μ plane. The resulting multiple complex branch points are found at the Roberge-Weiss values of Im μ, with Re μ defined by the values of Polyakov lines and colormagnetic confinement.