When does the product of two polynomials = $x^{k}$? Suppose $f$ and $g$ are are two polynomials with complex coefficents (i.e $f,g \in \mathbb{C}[x]$).
Let $m$ be the order of $f$ and let $n$ be the order of $g$.
Are there some general conditions where  
$fg= \alpha x^{n+m}$
for some non-zero $\alpha \in \mathbb{C}$ 

Yes, the intuitively evident ones: all other terms in $f$ and $g$ must vanish.  To see this, note that the product of the constant terms of $f$ and $g$ equals the constant term of $fg$, which is zero, whence at least one of these polynomials is multiple of $x$.  Without any loss of generality assume it is $f$.  Then
$$fg = x\, \left(\frac{f}{x}\right) g,$$
implying $(f/x) g$ is a multiple of $x^{n+m-1}$.  By induction this reduces us to the case $n+m=0$, which is trivial (because $f$ and $g$ then have no other terms). QED.
