What does it mean if the standard Hermitian form of complex two vectors is purely imaginary? If $v,w \in \mathbb{C}^n$, what does it mean geometrically for $\langle v , w \rangle$ to be purely imaginary? 

I'm not sure what it means geometrically, but it means $\| v + w \|^2 = \|v \|^2 + \|w^2\|$. 
$\langle v, w \rangle$ purely imaginary means that $\overline{\langle v, w \rangle } = - \langle v, w \rangle$.  However, we always have $\overline{\langle v, w \rangle } = \langle w, v \rangle$.  This means $$ \langle w, v \rangle = - \langle v, w \rangle \implies \langle w, v \rangle  + \langle v, w \rangle = 0.$$
This implies that \begin{align*} \| v + w \|^2 &= \langle v + w, v + w \rangle \\
&= \langle v, v\rangle + \langle v,w\rangle + \langle w, v \rangle + \langle w, w \rangle \\
&= \| v \|^2 + \| w \|^2.
\end{align*}
