Let $u : U \rightarrow \mathbb{R}$, where $U \subset \mathbb{C}^{n}$ be a strictly plurisubharmonic smooth function and consider its complex hesse matrix $Hess^{\mathbb{C}}(u)$. Furthermore consider a diffeomorphism $\varphi : V \subset \mathbb{C}^{n} \rightarrow U$. And again consider the complex hesse matrix of $u' := u \circ \varphi : V \rightarrow \mathbb{R}$, $Hess^{\mathbb{C}}(u')$. My question is: How are these two matrices related to each other under this change by the diffeomorphism? Or is there any literature ? 



hesse matrix under diffeomorphism
I guess your diffeomorphism is a biholomorphism, or at least an holomorphic map, otherwise there is no hope to say much about it.


Then the best way to look at these things is using differential forms.
Indeed, $\imath \partial \bar{\partial} u =\imath \sum\_{i,j} \frac{\partial^2 u}{\partial z\_i \partial \bar z\_j} dz\_i \wedge d\bar z\_j$. 
As is well known, as $\varphi$ is holomorphic (not necessarily bijective), $\varphi^\* $ commutes with $\partial$ and $\bar \partial$. 
Therefore $\imath \partial \bar{\partial} u' = \varphi^\* \imath \partial \bar{\partial} u$, which you can translate in terms of matrices then, if you want to:


$$Hess(u')\_a(\xi)= \underset{j,k,l,m}{\sum} \frac{\partial^2 u(\varphi(a))}{\partial z\_l \partial \bar z\_m} \frac{\partial \varphi\_l(a)}{\partial z\_j } \xi\_j \overline{\frac{\partial\varphi\_m(a)}{\partial z\_k } \xi\_k} $$
if $\varphi= (\varphi\_1, \ldots, \varphi\_n)$, and $\xi$ is any tangent vector at $a$.


