It's an unfortunate tendency in textbooks on Lie algebras to assume that the base field is $\mathbb{C}$ even though virtually everything done in the classical structure theory and finite dimensional representation theory remains valid over an arbitrary algebraically closed field of characteristic 0. For the theory of affine Lie algebras the book by Kac is an essential foundational source, but a possibly more readable textbook account of his ideas was given by Roger Carter in his 2005 Cambridge University Press book Lie Algebras of Finite and Affine Type. Though Carter also tends to work by default just over the complex field, it's reasonably clear throughout that the arguments used are purely algebraic.
Concerning the introduction of twisted affine algebras (as in Carter's Chapter 18), there may be slightly different viewpoints. But for example Carter initially classifies generalized Cartan matrices and related data, then shows how to realize suitable Lie algebras having that data. In the process only the graph automorphisms play an essential role.
Maybe there are good alternatives to such books(?), but it's worthwhile anyway to consult Carter's book in order to clarify other issues you may encounter when reading denser parts of the Kac book. In principle they are both talking about the same objects.
ADDED: To respond in part to Angelo's comments, I'm not convinced that the limited context here requires any use of complex exponentials. First, it's generally agreed that the finite dimensional simple Lie algebras over $\mathbb{C}$ can be studied adequately over any algebraically closed field of characteristic 0.
But Kac needs to deal with the automorphism group of this Lie algebra as well. That can be studied from two viewpoints: complex Lie groups or algebraic groups over an algebraically closed field of characteristic 0. The automorphism group itself is essentially the same (up to base change) in either case, but it may be more convenient to construct "diagonal" automorphisms by direct use of exponentiation applied to elements of a Cartan subalgebra of the Lie algebra. From the algebraic viewpoint this is where it gets tricky, whereas automorphisms coming from root vectors can be studied using polynomial versions of exponential series. There was work by Chevalley and then by Steinberg sorting out such questions. When Kac works with automorphisms of finite order, he probably finds it more natural to work over $\mathbb{C}$ rather than get there less directly. But I still think it's a matter of taste and not related essentially to the outcome for affine Lie algebras. (That said, I've been away from this literature quite a while.)