With non-Newtonian fluids, the viscosity is no longer constant. Depending on the flow conditions (and the material of course), the viscosity will decrease as the shear rate increases (shear thinning) or it will increase (shear thickening). In some cases as well, there is a time dependency - the viscosity at a given flow condition can decrease over time (thixotropy) or increase (rheopexy).
What I've just describer however, are ideal models that don't exist in reality. As far as I am aware, a purely shear-thinning material does not exist. Sure, most thermoplastics do show shear-thinning behavior, but it is over a range of shear rates, rather than ALL shear rates. (At low shear rates, the viscosity becomes constant and is given the name of zero-shear viscosity, and a some materials will also show a constant, high shear rate viscosity). The same is true for shear thickening materials - such behavior is observed over a range of shear rates and not in other subranges. And it becomes a bigger mess when time-dependent behavior is added. I've never seen a thixotropic material that wasn't also shear-thinning.
It would be a great help for educational purposes if we could find such materials that show only these isolated ideals of non-Newtonian behavior, particularly with the time dependent phenomenon. I think all educators have students that struggle with the difference between shear-thinning and thixotropy. A material showing only thixotropic behavior would be a great aid in demonstrating the difference.
Lastly, there is one set of fluids I am aware of that are able to isolate one subset of non-Newtonian behavior and those are Boger fluids - they have a constant viscosity like a Newtonian liquid, but show large amounts of elasticity.