In this question, a simple formula was provided for the geodesic (and as such the exponential map) given a starting point $p$ and an initial direction $\mathbf v$ on the surface of the real unit $n$-sphere embedded in real euclidean space $\mathbb R^{n+1}$:
$$\gamma(t)=\cos(||\mathbf v||t)\mathbf p +\sin(||\mathbf v||t)\frac{\mathbf v}{||\mathbf v||}$$
Is there any equivalent for this formula for the complex unit $n$-sphere embedded in complex euclidean space $\mathbb C^{n+1}$? Does the original formula still work? Through some quick numeric investigation, the latter does not seem to be the case.
My definition of the complex unit $n$-sphere is the set of vectors in $\mathbb C^{n+1}$ which admit to $\sqrt {x^\dagger x} = 1$ (so their "euclidian norm" should equal 1).
Background of the question: I am trying to apply techniques from optimizing on a regular $n$-sphere (which only require an exponential map) to the complex $n$-sphere. My local point, the gradient of the target property at that point and its tangent projection on the tangent bundle at the point are all given as vectors with complex entries. Just using the same geodesic (as pointed out above) does not cut it for some reason.
As pointed out the in the comments, $\mathbb C^n$ is isomorph to $\mathbb R^{2n}$ which enables use of the geodesic from the question in that space.
(This answer was posted for the sole reason of marking this question as answered. If there is another way to do so without posting an official answer please let me know).