Is there a $\mathbb{C}$ vector space structure over $\mathbb{R}$ with scalar multiplication as defined? To find is a $\mathbb{C}$ vector space structure over $\mathbb{R}$ with addition and scalar multiplication such that scalar multiplication has the following property:

$\cdot : \mathbb{C} \times \mathbb{R} \rightarrow \mathbb{R}$ s.t.

$\cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is a usual multiplication
 in $\mathbb{R}$
No. For any vector space $V$, any nonzero vector $v$, and any scalars $k$ and $l$, $kv=lv$ implies that $k=l$. Here, $kv$ already takes all possible values when $k$ ranges over $\Bbb R$, so there's nothing for (say) $iv$ to be.
This supposed vector space would be $1-$dimensional. Let $\{a\}$ be a basis. Then $i\cdot a = r =\frac{r}{a}\cdot a , \ \ r \in \mathbb{R}$ ↯.