Let $M$ be a complex manifold. I have seen the bundle of complex valued $k$-forms defined as $\Lambda^k(M) \otimes \mathbb{C}$. What this means is that if $V := T_xM$ for some $x \in M$, then the fiber at $x$ is $\Lambda^k( \hom_\mathbb R (V, \mathbb R) ) \otimes_\mathbb R \mathbb C$. What I know is that: $$ \Lambda^k( \hom_\mathbb R (V, \mathbb R) ) \otimes_\mathbb R \mathbb C \simeq_\mathbb R \hom_\mathbb R (\Lambda^k (V), \mathbb C) $$ and the $\mathbb R$ vector space on the right hand side is isomorphic to the $\mathbb R$-vector space of $\mathbb R$-multilinear alternate maps from $V^k$ to $\mathbb C$.
However, I have also seen that for example $(1,0)$-forms are characterized by being $0$ on $(1,0)$-vectors, and $(1,0)$ vectors live inside $V^\mathbb C:= V \otimes_\mathbb R \mathbb C$. So it must be the case that these complex valued forms take as arguments elements from $V^\mathbb C$.
So the question is, how do I see elements in $\hom_\mathbb R (\Lambda^k(V), \mathbb C)$ as maps taking as arguments elements from $(V^\mathbb C)^k$?
The only two options that I can logically think of to make my forms take arguments from $V^\mathbb C$ are either $$ \Lambda^k( \hom_\mathbb R (V^\mathbb C, \mathbb R))\otimes_\mathbb R \mathbb C \simeq_\mathbb R \hom_\mathbb R(\Lambda^k(V^\mathbb C), \mathbb C), $$ i.e. $\mathbb R$-multilinear maps from $(V^\mathbb C)^k$ to $\mathbb C$ (seen as an $\mathbb R$-vector space), or $$ \Lambda^k( \hom_\mathbb C(V^\mathbb C, \mathbb C)) \simeq_\mathbb C \hom_\mathbb C(\Lambda^k(V^\mathbb C), \mathbb C) $$ i.e. $\mathbb C$-multilinear maps from $(V^\mathbb C)^k$ to $\mathbb C$ (seen as a $\mathbb C$-vector space).
It is not clear to me whether one of these two latter vector spaces is isomorphic with $\hom_\mathbb R(\Lambda^k(V), \mathbb C)$ and if so, over which field - over $\mathbb R$ or over $\mathbb C$. This is why I don't know whether my question is one of convention or of linear algebra.
In short, when we say complex-valued forms, do we mean $\hom_\mathbb C(\Lambda^k(V^\mathbb C), \mathbb C)$ as a $\mathbb C$ vector space? Do we mean $\hom_\mathbb R(\Lambda^k(V), \mathbb C)$ as an $\mathbb R$-vector space or, if not, as a $\mathbb C$-vector space etc.? And are some of these in fact equivalent?