Advanced supercapacitor electrodes require the development of materials with dense redox sites embedded into conductive and porous skeletons. Two-dimensional (2D) conjugated metal-organic frameworks (c-MOFs) are attractive supercapacitor electrode materials due to their high intrinsic electrical conductivities, large specific surface areas, and quasi-one-dimensional aligned pore arrays. However, the reported 2D c-MOFs still suffer from unsatisfying specific capacitances and narrow potential windows because large and redox-inactive building blocks lead to low redox-site densities of 2D c-MOFs. Herein, we demonstrate the dual-redox-site 2D c-MOFs with copper phthalocyanine building blocks linked by metal-bis(iminobenzosemiquinoid) (M2[CuPc(NH)8], M = Ni or Cu), which depict both large specific capacitances and wide potential windows. Experimental results accompanied by theoretical calculations verify that phthalocyanine monomers and metal-bis(iminobenzosemiquinoid) linkages serve as respective redox sites for pseudocapacitive cation (Na+) and anion (SO42-) storage, enabling the continuous Faradaic reactions of M2[CuPc(NH)8] occurring in a large potential window of -0.8 to 0.8 V vs Ag/AgCl (3 M KCl). The decent conductivity (0.8 S m-1) and high active-site density further endow the Ni2[CuPc(NH)8] with a remarkable specific capacitance (400 F g-1 at 0.5 A g-1) and excellent rate capability (183 F g-1 at 20 A g-1). Quasi-solid-state symmetric supercapacitors are further assembled to demonstrate the practical application of Ni2[CuPc(NH)8] electrode, which deliver a state-of-the-art energy density of 51.6 Wh kg-1 and a peak power density of 32.1 kW kg-1.