Theorem (Chevalley)
The functor that takes linear algebraic groups $G$ to their $\mathbb{R}$-points $G(\mathbb{R})$ constitutes an equivalence of categories between compact Lie groups and $\mathbb{R}$-aniosotropic reductive algebraic groups over $\mathbb{R}$ all whose connected components have $\mathbb{R}$-points.
For $G$ as in this equivalence, then the complex Lie group $G(\mathbb{C})$ is the complexification of $G(\mathbb{R})$.
(from (Conrad 10))
James Milne, Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups (web)
Lie algebras, algebraic groups and Lie groups (pdf)
A. L.Onishchik, E. B. Vinberg, Lie groups and algebraic groups, Springer 1990
Richard Pink, Compact subgroups of linear algebraic groups (pdf)
Brian Conrad, MO comment 2010