$$\vert z+w\vert^2+\vert z-w\vert^2=(z+w)\cdot(\overline{z+w})+(z-w)\cdot(\overline{z-w})$$$$=(z+w)\cdot(\overline z+\overline w)+(z-w)\cdot(\overline z-\overline w)$$$$=(z\cdot\overline z+z\cdot\overline w+w\cdot\overline z+w\cdot\overline w)+(z\cdot\overline z-z\cdot\overline w-w\cdot\overline z+w\cdot\overline w)$$$$=2z\cdot\overline z+2w\cdot\overline w=2\vert z\vert^2+2\vert w\vert^2.$$Gruß