Can the equation for the principal curvatures, $k^2 - 2Hk + K = 0$ (where H is equal to the mean curvature and K is equal to the Gaussian curvature), ever have complex roots?
In other words, where the roots are $k_1 = H + \sqrt{H^2 - K}$ and $k_2 = H - \sqrt{H^2 - K}$, will $H^2 - K$ ever be less than zero for some real-world object?
If by "real-world" you mean a regular surface embedded in Euclidean $3$-space, then "no": The shape operator of a regular suface $S$ at a point $p$ is a symmetric operator on $T_{p}S$, so its eigenvalues $k_{1}$ and $k_{2}$ are real, and $$ H^{2} - K = \bigl[\tfrac{1}{2}(k_{1} + k_{2})\bigr]^{2} - k_{1} k_{2} = \bigl[\tfrac{1}{2}(k_{1} - k_{2})\bigr]^{2} \geq 0. $$