This is a statement made in Lectures in Kahler Geometry by Moroianu 5.4 for comparison between levi-civita connection and chern connection.
A product $iX$ for some $X\in T_CM$ where $T_CM$ is the complexified tangent bundle of almost complex structure $(M,J)$ with $iX$ being identified with $JX$.
Q: Where is this identification coming from? Is this valid?
The reason to ask this question is that the author is trying to prove the following lemma.
$(M,h,J)$ is a hermitian manifold. For every $Y\in\Gamma(T_CM)$, $\bar{\partial}$ operator as $T_CM$valued $(0,1)$ form is given by $\bar{\partial}^\nabla Y(X)=\frac{1}{2}(\nabla_XY+J\nabla_{JX}Y-J(\nabla_YJ)X)$ where $\nabla$ denotes levi-civita connection of any $h$ hermitian metric on $M$.
The proof involves showing $\bar{\partial}^\nabla$ obeying leibniz rule.
One of the critical step involves $f\in C^{\infty}(M),\frac{1}{2}(X(f))Y+JX(f)JY)=\frac{1}{2}((X+iJX)(f))Y$ where one identifies $JY=iY$ and pulls out $i$.
Q: How do I see this identification here?