Bonjour/bonsoir à toutes et à tous.

This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric Banach spaces (over the real or the complex field).

Question 1. Do $\mathbf{X}$ and $\mathbf{Y}$ need to share the same Hamel dimension?

The answer is clearly yes if the scalar field is the real numbers, since then any surjective isometry $\mathbf{X} \to \mathbf{Y}$ is, a fortiori, an affine transformation $X \to Y$ (via the Mazur-Ulam theorem). But what about the complex case?

Question 2. Do $\mathbf{X}$ and $\mathbf{Y}$ need to share the same (extended) Schauder dimension as defined in J. W. Evans and R. A. Tapia, Hamel Versus Schauder Dimension, The American Mathematical Monthly, Vol. 77, No. 4 (Apr., 1970), pp. 385-388 (click)?

Notes. 1) I'm using the term isometry to refer, herein, to both linear and non-linear isometries.