Does $〈Ax,y⟩=〈x,A^*y⟩$ hold for any inner product? Consider $\Bbb C^n$. Let $A^*$ be the conjugate transpose of matrix $A$, I see in many materials $A^*$ is also called the adjoint of $A$. I am now confused by that does $〈Ax,y⟩=〈x,A^*y⟩$ hold just for the standard inner product (whose proof is simple), or any inner product? Thanks!
It does not hold for every inner product. The analogous situation in the real plane illustrates this. Consider
$$
H = \pmatrix{2 &amp; 0 \\ 0 &amp; 1}
$$
and the inner product $x \cdot y = x^t H y$. 
Let 
$$
A = \pmatrix{0 &amp; 1 \\ 1 &amp; 0}
$$
$x =  \pmatrix{1\\0}$ and $y =  \pmatrix{0\\1}$. 

Then $Ax \cdot y = 1$, but $x \cdot A^t y = 2$. 

Since the real plane is a subspace of the complex plane, this suffices as a counterexample in $\Bbb C^2$ as well (at least if you extend the definition of the dot product by writing $x \cdot y = x^t H \bar{y}$).
$〈Ax,y⟩=〈x,A^*y⟩ $ this holds for any inner product ! when you assume $A^*$ is adjoint of $A$ in space X equipped with inner product of $〈.,.⟩$

This is how adjoint operators  defined in finite dimension.