Magnetically charged particle or monopole theoretically predicted by Dirac almost a century ago still remains elusive, because it has not been revealed experimentally. So, should we continue to hunt for it? Instead, this work aims to establish charge symmetry in Maxwell electrodynamics and to explain electric charge quantization without attracting the monopole concept. An alternative approach is proposed which leads to new equations which are considered as counterparts of Maxwell equations. It is shown that these new equations can describe "rotating" electric and magnetic fields, with the elementary magnetic moment and its temporal change as sources for these fields, respectively. The charge quantization problem is considered closely to Dirac's approach; however, an alternative solution of the problem is shown by changing topology of the surface through which the magnetic flux emanates. This way leads to introduction of a quantized magnetic moment along the z-axis due to spin motion of the charged particle, which is connected to the charge value through fundamental constants. The suggested approach stimulates those physicists who try to deeper understand fundamentals of our world.