If you try stepwise elimination of either fx or fy with respect to either x or y, you will get a solution involving the roots of a polynomial of order 12. With each of them being order 12, MATLAB has to work through 144 different roots -- but it cannot express them in full because there is not typically any closed form expression for any order above order 4. So this is not an easy equation to solve.
Unfortunately proceeding numerically does not work well in this case: the loss of precision that goes along with working with floating point numbers can result in somewhat negative values being calculated from fx^2+fy^2 even though algebraically that should not be possible.
You appear to be trying to calculate all of the extrema and inflections of f(x,y), including the complex ones.
If you are just looking for the global minima, then it is at about -4.34915949152492498, 1.72539721154691694
