Consider the 3-sphere $S^3$. If I look it as a embedded in $\mathbb{R}^4$, then the tangent space in a point $x \in \mathbb{R}^4$ is isomorphic to $\mathbb{R}^3$. But, if I consider
$$S^3 = \{(z,w) \in \mathbb{C}^2| \ |z|^2 + |w|^2 = 1\},$$ given a point $a \in \mathbb{C}^2$, how the tangent $T_a S^3$ looks like?
I don't know if this question makes sense because $S^3$ is a real manifold. I saw questions related, but I didn't find the answer.
Appreciate.
The tangent space is the real three-dimensional affine subspace of $\mathbf{C}^{2}$ passing through $a$ and real-orthogonal to $a$, just as it would be for the unit sphere in $\mathbf{R}^{4}$. In a bit more detail, the complex line spanned by $a$ intersects $S^{3}$ in a Hopf circle $C_{a}$. The tangent space to $S^{3}$ at $a$ is spanned by the real tangent line to $C_{a}$ at $a$ and any complex line other than the line spanned by $a$.