This study investigates whether the electromagnetic and gravitoelectromagnetic energy conservation equations are obtained together by using octonion algebra or not. Maxwell and Maxwell-like equations for linear gravity with magnetic monopole terms are used in the SI unit system. A new complex octonionic field term is suggested for the first time. The complex octonionic source equation is then obtained. Finally, Poynting theorems for both electromagnetism and gravitoelectromagnetism are de fined for the first time by using higher dimensional algebra.