Let $H$ be a complex vector space and $<,> :H \times H \to \mathbb{C}$ a map having the following properties:
Linear in the first argument
Conjugate (or anti-)linear in the second argument
Positive definite, i.e., $<v,v>$ is always nonnegative and in particular, positive if and only if $v \neq 0$
Then, what is the problem with this $<,>$?
The only thing that is missing is that I did NOT impose $<v,w>=\overline{<w,v>}$. However I cannot find anything that keeps me from proclaiming that this is an inner product.
Could anyone please elaborate?
There is nothing wrong. You can derive $\langle v,w\rangle=\overline{\langle w,v\rangle}$ from your assumptions:
From expanding $\langle v+w,v+w\rangle-\langle v,v\rangle-\langle w,w\rangle\in\mathbb{R}$ we have $\langle v,w\rangle+\langle w,v\rangle\in\mathbb{R}$, so $$\operatorname{Im}\langle v,w\rangle=-\operatorname{Im}\langle w,v\rangle.$$ Replacing $w$ by $iw$, we have $$ \operatorname{Im}\langle v,iw\rangle=-\operatorname{Im}\langle iw,v\rangle. $$ But LHS is $\operatorname{Im}(-i)\langle v,w\rangle=-\operatorname{Re}\langle v,w\rangle$ and RHS is $-\operatorname{Im}i\langle w,v\rangle=-\operatorname{Re}\langle w,v\rangle$. So $\langle v,w\rangle$ and $\langle w,v\rangle$ have equal real parts and opposite imaginary parts, i.e., they are complex conjugate.