Rigid motion which is not a linear transformation From Linear Algebra Done Wrong:

Give an example of a rigid motion in $\mathbb{C}^{n}$, $T(\mathbf{0}) = \mathbf{0}$, which is not a linear transformation.

where a rigid motion was defined as a transformation $f : V \to V$ on an inner product space $V$ such that
\begin{align}
\left\Vert f(\mathbf{x}) - f(\mathbf{y}) \right\Vert = \left\Vert \mathbf{x} - \mathbf{y} \right\Vert && \text{$\forall\mathbf{x},\mathbf{y} \in V$}.
\end{align}
I've so far found examples of transformations which are linear and not rigid (such as $T(z) = \alpha \cdot z$, where $\alpha \in \mathbb{C}$, $z = x + iy$) and transformations that are neither linear nor rigid (such as $T(x + iy) = 2x - y + iy$). Translations ($T(z) = z + a$, $a \in \mathbb{C}$) appear to be ruled out by the condition that $T(\mathbf{0}) = \mathbf{0}$. Beyond this, I'm not sure how to proceed. Any hint would be greatly appreciated.

Hint: Look for a map that is linear with respect to coefficients from $\Bbb R$, but not linear with respect to coefficients from $\Bbb C$.
