What is known on the triviality of the complexified tangent bundle of the spheres? Are they always trivial?
Here is my progress. If we are just looking at any rank n complex vector bundle, then this would correspond to looking at $S^n \to BU(n)$ or equivalently $\pi_n(BU(n)) \cong \pi_{n-1}(U(n)) \cong \pi_{n-1}(U(n-1))$. However $\pi_{n-1}(U(n-1))$ can still be nonzero for even n as it alternates between 0 and $\mathbb{Z}$. So we are left with even dimensional spheres. Of course this does not use that it is the tangent bundle (not any other complex rank n vector bundle), but I could not figure out how to integrate this in the argument. Of course in the cases where tangent bundle of sphere is parallelizable (dim=1,3 and 7), the complexification will also be trivial, but these are already covered by above argument.
When $n=2$, Triviality of complexified tangent bundle of a closed surface shows that complexification is trivial. What about even dimensional spheres for $2n \geq 4$?
Here is an attempt with characteristic classes: $TS^n \oplus \epsilon_{S^n}^1\cong TS^n \oplus \nu=\epsilon_{S^n}^{n+1}$ where $\nu$ stands for the trivial normal bundle, now we can look at Pontrjagin classes of $TS^n$, those are obtained after complexifying the tangent bundle hence may carry information on nontriviality of complexified tangent bundle. However $p(TS^n)=p(TS^n \oplus \epsilon_{S^n}^1)=p(\epsilon_{S^n}^{n+1})$ by the stable property of Pontrjagin classes hence we do not get any nontrivial info.