Photonic conical dispersion has been found in either transverse magnetic or transverse electric polarization, and the predominant zero-refractive-index behavior in a two-dimensional photonic crystal is polarization-dependent. Here, we show that two-dimensional photonic hypercrystals can be designed that exhibit polarization independent conical dispersion at the Brillouin zone center, as two sets of triply-degenerate point for each polarization are accidentally at the same Dirac frequency. Such photonic hypercrystals consist of periodic dielectric cylinders embedded in elliptic metamaterials, and can be viewed as full-polarized near zero-refractive-index materials around Dirac frequency by using average eigen-field evaluation. Numerical simulations including directional emissions and invisibility cloak are employed to further demonstrate the double-zero-index characteristics for both polarizations in the photonic hypercrystals.