From the perspective of functional analysis, why is it that we so often focus on vector spaces (Hilbert, Banach, whatever) over $\mathbb{C}$ instead of $\mathbb{R}$? Many of these objects obviously have natural real and complex versions, but I don't yet understand why we spend so much time on complex Hilbert spaces or $*$-algebras or whatever, when my naive guess would be that the theory is simpler over $\mathbb{R}$ and we could then extrapolate from there to the complex setting.
I assume that algebraic completeness factors into the equation, but while I can see the importance of working over $\mathbb{C}$ in the finite-dimensional world of linear algebra, where we have niceties like characteristic polynomials, polynomials seem themselves rather "finitary" in nature, and seem to often kinda disappear once we move to the infinite-dimensional setting. Where in the finite-dimensional setting we can say, "every matrix has a characteristic polynomial, so it must have at least one eigenvalue over $\mathbb{C}$ because the characteristic polynomial will split", once we move to even a separable space like $\ell^p(\mathbb{N})$, we can lose that luxury. My assumption would be that when polynomials become scarce, the ability to solve them because less vital.
One counterexample to this I could think of would be the density of $1$-dimensional representations in $L^1(G)$ for a compact group $G$. If I want to study $S^1$, I have to study the continuous homomorphisms $S^1 \to \mathbb{T} : = \{ z \in \mathbb{C} : |z| = 1 \}$. I can't get away with studying just the homomorphisms into the real part of $\mathbb{T}$ because that'd only leave me with the trivial homomorphism. I can still study real-valued functions on $S^1$ with Fourier analysis, but I have to drop down from the complex world to the real, not the other way around. I am starting with $\left\{ e^{2 \pi i k t} : k \in \mathbb{Z} \right\}$ and coming back to the real-valued functions on $S^1$ from there.
What are some other examples of results or concepts in analysis where the complex view of the field is indispensable when compared to the merely real?