The static properties of an anharmonic potential model for planar two-electron quantum dots are investigated using a method that allows for the exact representation of the matrix elements, including the full Coulombic electron-electron interaction. A quartic perturbation of the harmonic confining potential in combination with the interparticle Coulomb interaction affects the spectral properties of the system considerably as it implies total loss of separability in the dynamics. Consequently, the classical phase space is mixed regular-chaotic and standard spectral measures of quantum chaos indicate an intermediate degree of complexity. Apart from the prompt transition from a regular to a moderately chaotic regime for weak quartic perturbation, the complexity of the system appears to be insensitive to the strength of the quartic potential.