Let $\pi:E\to M$ be a complex bundle, where $M$ is a $n$-dimensional smooth manifold. We can consider the bundle $\text{Hom}(E,E)=:\text{End}(E)$ of endomorphism of $E$, with fibers
$$\text{End}(E)_{p}=\{\Bbb C\text{ -linear maps } E_{p}\to E_{p} \}$$
where $E_{p}$ is the fiber at $p\in M$ of the bundle $\pi:E\to M$. Assume that this bundle $\pi:E\to M$ is equipped with a fiber-wise hermitian inner product $\langle\cdot,\cdot\rangle$. We can consider the subbundle $\mathcal{H}(E)$ (of the bundle $\text{End}(E)$) of hermitian maps from $E$ to $E$, whose fibers are:
$$\mathcal{H}(E)_{p}:=\{H:E_{p}\to E_{p}\mid H=-H^{\ast}\}$$
where $H^{\ast}$ is the adjoint of $H$. An hermitian map $H:V\to V$ is a $\Bbb C$-linear map such that $H=-H^{\ast}$, which means: $\langle Hu,v\rangle+\langle u,Hv\rangle=0$ for all $u,v\in V$.
In this link (page 5, below definition 3.5), they affirm that if $\pi:E\to M$ is a complex line bundle, we have:
$$\mathcal{H}(E)\simeq M\times i\Bbb R$$
where I presume it means that they are diffeomorphic. How do they arrive to this conclusion? Is this related to almost complex structure?
EDIT: I realised they probably mean that they are isomorphic as vector bundles, which is the only way for this to make sense. I am working on it but open to suggestions on how to prove it.
What I have considered so far (but doesn't lead me anywhere):
Intuitively, on a purely linear-algebraic point of view, I understand that an hermitian map is "purely imaginary", in some sense (making the analogy with the fact that if $z$ is a complex number such that $z=-\overline{z}$, it implies that $z\in i\Bbb R$). As we have a complex line bundle, each fiber is isomorphic to a complex vector space $V$ of dimension $1$ (i.e. they are isomorphic to $\Bbb C$) and are complex vector spaces themselves, although $E$ remains a real manifold.
By definition of the bundle $\pi:E\to M$, for any $p\in M$, there exists an open neighbourhood $U\ni p$ and a diffeomorphism
$$\phi_{U}:\pi^{-1}(U)\to U\times \Bbb C$$
such that $\phi_{U}$ maps $E_{p}$ onto $\{p\}\times \Bbb C$.
For the bundle $\pi':\text{End}(E)\to M$, as $E_{p}$ is a one-dimensional complex vector space for all $p\in M$, $\text{End}(E)_{p}$ is also a one-dimensional complex vector space for all $p\in M$.