The conjugate symmetry axiom for the inner product is as follows:
$$\overline{\langle x,y\rangle } = \langle y,x\rangle$$
It is my understanding that the conjugate is there so that the norm of a vector can be defined, but I do not understand why it is also necessary to swap the order, why not define it like this:
$$\overline{\langle x,y\rangle } = \langle x,y\rangle$$
I can understand why that swapping the order may be useful if both $x$, and $y$ are real, since this allows the inner product to behave like the dot product, but I do not understand how complex values of $x$ and $y$ fit into this.
For example, is the following true when both $x$ and $y$ are complex?
$$\langle x,y\rangle =\langle y,x\rangle$$