Although a large number of fluids used in hydraulic fracturing practice exhibit a shear thinning behaviour, little is known on the impact of such a complex fluid rheology on the propagation of a hydraulic fracture. We focus our investigation on the configuration of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We allow for the occurrence of a region without fluid of a-priori unknown length at the fracture tip. We use the Carreau rheological model in order to properly account for the shear thinning of fracturing fluid between the low and large shear rates Newtonian limits. We solve this problem numerically combining a Gauss-Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the width-averaged lubrication flow. This yields in a system of non-linear equations for the fluid pressure in the filled region of the fracture and the extent of the fluid lag region near the fracture tip. We show that for a Carreau rheology, the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear thinning index and the amplitude of the shear thinning behaviour of the fluid (captured by the ratio of the high and low shear rate viscosities). The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are obtained analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate the response obtained with simpler rheological models (powerlaw, Ellis). In most cases, the power-law model does not accurately match the predictions obtained with a Carreau rheology. In the zero lag limit, the Ellis model properly reproduces the results of a Carreau rheology, albeit only for a dimensionless transition shear stress below a critical dimensionless transition shear stress whose expression is given analytically as function of the shear thinning index and magnitude.