It is one way of getting back from 8 dimensions to a real number via a multiplication.

  In a coordinate representation, the complex conjugate of an octonion, 
                A :=  _m=0S⁷  a_m i_m  =  a₀ i₀  +  _j=1S⁷  a_j i_j , 
may be defines as 
                A* :=  _m=0S⁷  a_m i_m*   =  a₀ i₀  -  _j=1S⁷  a_j i_j ,

where i_m  ( m = 0,....7 ) are basal elements of octonions, O, and  i₀ = 1 is real (i₀* =  i₀  ) .  Then
                n(A) := AA* = A*A  =  a₀²  + _j=1S⁷  a_j²
is a real number, called the norm of A.
     In the case of complex numbers the  conjugate of  z := a + ib , is z* := a - ib , and is a reflection of z about the real number axis. It requires only a little imagination to visualize the conjugate of an octonion.
     As with complex numbers,     (A*)* = A  .
     If A and B are two octonions, then     ( A B )* = B* A*  .
 ( i.e. it is said that the conjugation is an anti automorphism ).
  One can find the inverse, A^-1, of a not zero octonion, A, from its conjugate :   AA* = n (A)  implies that
 

The conjugation is just one case of a more general concept of involution.