For a real valued vector $x(t)$ that depends on some parameter $t$ and when requiring that $$\left|x(t)\right| = \mathrm{const}$$ it is easy to show that $$\dot{x}(t) \perp x(t)$$ where the dot denotes the derivative with respect to $t$. For a derivation see for example this math.stackexchange question. The basic idea goes like this:
$$x(t)\cdot\dot{x}(t) = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(x(t)\cdot x(t)) = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \left|x(t)\right|^2 = 0$$
My question is, can this be generalized to complex vectors $x \in \mathbb{C}^n$? Additionally, can you provide a textbook that deals with this problem?
We have $|x|^2=x^H x$, where $x^H$ is the conjugate transpose of $x$. Then, constant norm implies that
$$0=\frac{\mathrm{d}}{\mathrm{d}t}\left(x^H x\right)=\dot{x}^Hx+{x}^H\dot{x}=2\Re\left(x^H\dot{x}\right)$$
because $\dot{x}^Hx$ is the conjugate of ${x}^H\dot{x}$. So the equivalent condition for complex-valued $x(t)$ is
$$\Re\left(x^H\dot{x}\right)=0.$$
For example, $x=e^{it}$ is a one-component vector with constant norm, and we have
$$\Re\left(x^H\dot{x}\right)=\Re\left(e^{-it} i e^{it}\right)=\Re\left(i\right)=0.$$