Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $TM \otimes\mathbb{C}$ as what there exists and how can we do the differentiation from the sections of this bundle by using the Levi-Civita connection of the metric $g$?

I mean If $X, Y$ are two sections of $TM \otimes\mathbb{C}$ then the inner product of $X, Y$ how can be defined by using the metric $g$?

Moreover, how can we differentiate from $X$ along $Y$ in a natural way by using the Levi-Civita connection of $g$?

I can not find a definition which describes such a metric and differentiation.