Let $V$ be a real vector space of even dimension $2n$, together with a symplectic pairing $\langle\cdot,\cdot\rangle:V\times V\rightarrow \mathbb R$. We are interested in the morphisms of $\mathbb R$-algebras $h:\mathbb C\rightarrow \operatorname{End}_{\mathbb R}(V)$ satisfying $\langle h(z)v,w\rangle =\langle v,h(\bar z)w\rangle$ for every $v,w\in V$, and such that $(v,w)\mapsto \langle v,h(i)w\rangle$ is a symmetric positive definite bilinear pairing on $V$.
Assume that we are given two such morphisms $h$ and $h'$, such that the $\mathbb C$-vector space structures on $V\otimes_{\mathbb R}\mathbb C$ induced by $h$ and $h'$ are isomorphic. Then $h$ and $h'$ are conjugate by an automorphism respecting the symplectic form pairing $\langle\cdot,\cdot\rangle$.
I need help with the above statement. Here is what I have done.
Consider $f$ an automorphism of $V\otimes_{\mathbb R}\mathbb C$ with the two different structures at the start and at the target. Essentially, $f$ is an $\mathbb R$-linear automorphism of $V$ such that $f(h(z)v)=h'(z)f(v)$ for all $z\in \mathbb C$ and $v\in V$. Thus, all I need to prove is $\langle f(v),f(w)\rangle=\langle v,w\rangle$ for all $v,w$.
I tried introducing $\mathbb R$-linear endomorphisms $\phi$ and $\psi$ of $V$ such that $\langle f(v),w\rangle = \langle \phi(v),h(i)w\rangle = \langle \psi(v),h'(i)w\rangle$, which are given by Riesz representation theorem applied to the non-degenerate pairings induced by $h(i)$ and $h'(i)$. I was able to prove relations $\psi = f\phi f^{-1}$ as well as $\psi=fh(i)=h'(i)f$ and $\phi = h(i)f$, but I still can't get to the desired identity.
I am still exploring different ways to tackle the problem. I would gladly help any hint/leads towards the solution.