In this paper, the natural convection of a complex fluid that contains both nanoparticles and gyrotactic microorganisms in a heated square cavity is considered. The Buongiorno model is applied to descirbe the nanofluid behaviours. Both the top and bottom horizontal walls of the cavity are adiabatic, and there is a temperature difference between the left and right vertical walls. The non-dimensional governing equations are obtained when the stream-vorticity formulation of function is used, which are solved by the recently developed robust Coiflet wavelet homotopy analysis method. A rigid verification for the solver is given. Besides, the effects of various physics parameters including the Rayleigh number, the buoyancy ratio parameter, the bioconvection Rayleigh number, the Prandtl number, the Brownian motion parameter, the thermophoresis parameter, the heat generation parameter, the Lewis number, the bioconvection Peclet number and the Schmidt number on this complicated natural convection are examined. It is known that natural convection is closely related to our daily life owing to its wide existence in nature and engineering applications. We believe that our work will make a significant contribution to a better understanding of the natural convection of a complex fluid in a cavity with suspensions of both inorganic nanoparticles and organic microorganisms.