There is no positive definite bilinear form on $V$ 
Let $V$ be a $\mathbb C$-vector space, $\dim V>1$. Then for any bilinear form $\phi$ there is $v\in V$ s.t. $\phi(v,v)=0$ (so there is no positiv definite bilinear form on $V$)

If one takes the usual multiplication then $\begin{pmatrix}i\\0\end{pmatrix}\cdot \begin{pmatrix}i\\0\end{pmatrix}=-1$, so for a particular bilinear form, positive definiteness fails, but I need some example which gives $0$ and for any bilinear form.

Let $v = (1,\dots,v_n)$.  We can write
$$
\phi(v,v) = p(v_1,v_2,\dots,v_n)
$$
That is, $\phi(v,v)$ gives us a polynomial on the entries of $v$.  
Suppose that $\phi$ is constant over $v$.  By bilinearity, $\phi = 0$.
Now, suppose that $\phi(v,v)$ is not constant.  Then there is some $v_i$ such that $p(v_1,v_2,\dots,v_n)$ is a non-constant polynomial on $v_i$.
For any fixed values of $v_j$ (for $j\neq i$), $p$ gives us a quadratic polynomial on $v_i$.  Thus, there is a value of $v_i$ that makes $p$ equal to zero.
In either case, there exists a $v$ such that $\phi(v,v) = 0$.
