Mazur-Ulam-like theorem for complex Hilbert spaces

Question

The Mazur-Ulam theorem doesn't hold for complex Hilbert spaces because antiunitary operators are origin preserving surjective isometries, but they aren't linear. Is it true, that every $f:H\to H$ surjective isometry of a complex Hilbert space $H$ has the form $$f(x)=f(0)+U(x)$$ where $U$ is unitary (hence linear) or antiunitary (hence antilinear)?

Answer 1

The map $(z, w)\in \mathbf C^2 \mapsto (z,\overline{w})\in \mathbf C^2$ is norm-preserving, but not linear nor antilinear.