A requirement is missed in your list. If $a,b \in \mathbb C$, it must also hold
$$\langle \alpha| a\beta + b\beta' \rangle = a\langle \alpha| \beta \rangle + b\langle \alpha|\beta' \rangle\:.$$
Regarding your question, consider the $2\times 2$ matrix $A:= diag(-2,1)$ and define in $\mathbb C^2$
$$\langle \vec{x}| \vec{y}\rangle := {\vec{x}^*}^t A \vec{y}$$
where $\vec{z} = (a,b)^t$ with $a,b \in \mathbb C$ and $\vec{z}^* = (a^*,b^*)^t$.
With this definition the first requirement is true (together the one I added), but the second is false because, if $\vec{x}=(1,1)^t$ you find
$$\langle \vec{x}| \vec{x} \rangle = -1\:.$$