Why complex vector space inner product is defined with conjugate symmetry?

If you don't use conjugate but simple symmetry it will produce a contradiction with the axiom X . X >= 0

assume X.Y = Y. X

X.X >=0

=> c1 X . c1 X >= 0 [where c1 is a complex number with an imaginary part]

=> c1 (c1 X . X) > = 0 by linearity

=> c1 (X . c1 X) > = 0 by symmetry

=> c1 . (c1 ( X. X) )>= 0 by linearity

=> (c1 . c1) . (X.X) > = 0 by associativity of complex numbers

which is a contradiction.

to see this, if you take c1 = i [square root of -1]

=> -1 (X.X) >= 0

=> (X.X) <= 0

cool, right?