Is complex analysis more "real" than real analysis? In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself had to be described by complex wave functions and there's no way to describe it using only real numbers.

In mathematics, in real analysis, there's examples like the function $f(x)=\frac{1}{1+x^2}$, and why this function does not have the "smoothtness" of the exponential function, polynomials, sine and cosine functions, why it has a radius of convergence equals 1 despite the fact that this function is infinitely differentiable. You can't see the reality of this function until you see it through the field of complex analysis, where you can observe that $f$ is not that smooth because it has 2 singularities in the complex plane.

I am just asking for examples like this such that when you see it in the narrow "window" of real analysis, you can't see the "reality" until you view it from the window of complex analysis. I am just starting to self learn complex analysis and I find it more natural than real analysis and it tells you the "truth" behind a lot of things.
Not all real degree $n$ polynomials have $n$ roots (counting multiplicity) because some of the roots are complex. In the real domain a matrix can have no eigenvalues, e.g. 2-dimensional rotation matrix, but any real matrix has a complex eigenvalue. These are manifestations of  $\mathbb{C}$ unlike $\mathbb{R}$ being algebraically closed, i.e. every polynomial equation having a solution. In the real domain $\sqrt{x}$ and $\ln{x}$ are only defined for positive $x$ because for negative $x$ the value is a complex number, and it is not unique. 

In the real domain exponents and trigonometric functions are completely different functions, but in the complex domain they are related by simple Euler formulas. The same goes for logarithms and inverse trigonometric functions. This is the main reason why identities for hyperbolic functions are almost the same as familiar trigonometric identities. Many definite integrals of functions that do not have elementary antiderivatives can be computed in elementary terms by extending the path of integration to the complex plane and using residues, e.g. $\int_0^\infty\frac{\ln x}{(1+x^2)^2}\,dx=-\frac{\pi}{4}$. More generally, integral and series representations of many real functions can be converted into each other because these functions extend into the complex plane and contour integrals there reduce to sums over residues. The Riemann zeta function is a typical beneficiary. These manifest another advantage of complex analysis over real one. Many commonly used real functions extend to holomorphic functions in the complex plane, and for holomorphic functions calculus tools are much stronger than for smooth ones, which is what real analysis mostly treats them as.

In the real domain ellipses and hyperbolas are different types of curves, but in the complex plane they are related by a rotation of axes, i.e. they are the 'same' (more precisely, we are looking at two different projections of the same complex curve). In a similar way spherical and hyperbolic geometries are related by a complex rotation. The Schrödinger equation of quantum mechanics and the heat equation of classical physics are also related by a complex rotation called Wick rotation. Path integral interpretation of quantum mechanical solutions can be made precise using this relation and the Feynman–Kac formula. 

Heaviside developed operational calculus for solving ordinary differential equations with constant coefficients by treating time derivative as a 'variable' $p$ and writing solutions in terms of symbolic 'functions' of it. It turned out that the magic worked because $p$ is in fact a complex variable, and Heaviside's symbolic solutions can be converted into real ones by taking the inverse Laplace transform, which is a contour integral in the complex plane.

Harmonic functions, solutions to the Laplace equation, have many nice analytic properties like being sums of convergent power series, attaining extrema at the boundary of their domains, being equal at any point to the average of values on any circle centered at it, etc. The underlying reason is that harmonic functions are exactly the real and imaginary parts of holomorphic functions. If the potential of a vector field is a harmonic function $\varphi$ then its flow lines are level curves of another harmonic function $\psi$, exactly the one that makes $\varphi+i\psi$ holomorphic. Solution formulas for the Dirichlet boundary problem for the Laplace equation in some special domains are reflections of the Cauchy integral fomula for holomorphic functions that works in 'any' domain.
Some things about complex analysis seem special when in fact those results are better viewed through the lens of vector calculus.

For instance, any holomorphic function can be viewed instead as a vector field with vanishing divergence and curl.  Consider the Cauchy integral theorem and Stokes' theorem:

$$\begin{align*}\text{Cauchy integral theorem: }&amp; \oint_C f(z) \, dz = 0, \quad \left(\frac{\partial f}{\partial \bar z} = 0 \right)\\
\text{Stokes' theorem: } &amp; \oint_C F \cdot d\ell = 0, \quad (\nabla \times F = 0) \\
\text{Stokes' theorem corollary: }&amp; \oint_C F \times d\ell = 0, \quad (\nabla \cdot F = 0) 
\end{align*}$$

Now, you may be tempted to say that the complex analysis version of this result is more elegant because it's one line instead of two.  You can package the vector calculus result into one line with the proper tools (clifford algebra) as well, though, and more importantly, when you properly identify the derivatives as exterior and interior derivatives, the result generalizes to arbitrarily high-dimensional real vector spaces.

Similarly, meromorphic functions correspond to vector fields with point sources, but that's usually where complex analysis stops, while vector calculus has the tools to treat vector fields with arbitrary sources.

That quantum mechanics is written in terms of complex numbers is, in my opinion, a pedagogical mistake.  It would be clearer to do quantum-style probability in terms of vectors in $\mathbb R^2$; of course, the math will ultimately come out the same, but I think the picture of things would be considerably cleaner.

Complex analysis also puts a lot of emphasis on complex differentiability, but as you can see from the above argument about the CIT and Stokes' theorem, it's actually differentiation with respect to $\bar z$ that relates very directly to vector calculus.  Complex differentiation is utterly insignificant when considering more general applications.