The author builds a linear complex algebra taking an arbitrary complex involution (i.e. an automorphism of the field \(\mathbb{C}\) of complex numbers of order two) instead of the ordinary conjugation. In the set \(M\) of all complex matrices an involution is defined as a map \(*:M\to M\) satisfying \((A^*)^* =A\) and \((AB)^* =B^*A^*\) in any possible case. It is shown that every involution of \(M\) is induced by a suitable complex involution (Th. 2.2). Other generalizations of the ordinary notions are presented: \(f\)-Hermitian matrices, unitary spaces with respect to the absolute value \(|z|_f= |f(z)|\) defined by an involution \(f\), generalized inverses and so on. This interesting paper contains other essential results. (Reviewer's remark: Since the identity and the complex conjugation are the only continuous automorphisms of \(\mathbb{C}\), the other \(\exp (\exp (\aleph_0))\) involutions of \(\mathbb{C}\) are non-measurable functions).