Suppose $L$ is a semi-simple Lie Algebra with dimension $d$. The complexification of the Lie Algebra over $\mathbb{R}$ is given by $$L_{\mathbb{C}} = L \otimes_{\mathbb{R}} \mathbb{C}$$
My question is: in general is there a way to determine the dimension of the complexification knowing the dimension of the original Lie Algebra?
On
"My question is: in general is there a way to determine the dimension of the complexification knowing the dimension of the original Lie Algebra?"
Answer: If $L$ is a real Lie algebra of dimension $d$, it follows $L*:=L\otimes_{\mathbb{R}}\mathbb{C}\cong \mathbb{C}^d$. Hence $L*$ has dimension $d$.
Comment: "could you expand on why L⊗RC≅Cd?"
Answer: The Lie algebra $L$ has a basis over the reals:
$$L \cong \mathbb{R}\{e_1,..,e_d\}$$
and it follows
$$L \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{R}\{e_1,..,e_d\}\otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C}^d.$$
This has nothing to do with Lie algebras. It's a general basic fact of linear algebra and scalar extensions that if $E \vert K$ is any field extension and $V$ is any vector space over $K$, then
$$\dim_E (E \otimes_K V) = \dim_K(V).$$
More precisely, if $\{v_i: i \in I\}$ is any $K$-basis of $V$, then $\{1 \otimes v_i : i\in I\}$ is an $E$-basis of $E \otimes_K V$. Any source which introduces tensor products should mention if not prove these facts.