Suppose $(x, y)$ is a product function on some vector space $X$ over the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We don't know that it's a scalar product, but we do know that it's 'sesquilinear'. My first question is, can 'sesquilinear' be interpreted as satisfying the equality $(\alpha x + \beta y, z) = \alpha(x, z) + \beta(y, z)$ for all $x, y, z \in X$ and $\alpha, \beta \in \mathbb{K}$?
Secondly, given sesquilinearity, can it be shown that if $(x, x) \in \mathbb{R}$ for all $x \in X$, then the bracket function has conjugate symmetry, i.e. for any $x, y \in X$, $(x, y) = \overline{(y, x)}$?
I've been trying to prove this by finding $(x+y, x+y)$, but I can't prove that $(x, y)$ and $(y, x)$ have the same real part.
(1) The term sesquilinearity is only used over $\mathbb{C}$, where it means additivity in each slot and additionally $$ (\alpha x , y) = \alpha (x,y) = (x,\bar \alpha y). $$ (You only wrote down the first equation)
(2) Over $\mathbb{C}$ it is true that any sequilinear form with $(x,x)\in\mathbb{R}$ (for all $x\in X)$ is hermitian, this follows from the polarization identity: $$4(x,y)=\sum_{k=0}^3 i^k(x+i^ky,x+i^ky)$$ Over $\mathbb{R}$ we always have $(x,x)\in\mathbb{R}$ and cannot conclude symmetry in general.