This is actually a really good question, because you're right to be suspicious about such a generalization having such nice properties.
Derivatives:
The definition for integrals and derivatives of complex-valued functions mirrors the definition of the real case quite a bit. Take a function $f : \mathbb{C} \to \mathbb{C}$. We define its derivative by
$$f'(\zeta) = \lim_{z \to \zeta} \frac{f(z) - f(\zeta)}{z - \zeta}$$
Note that $z,\zeta$ may be complex numbers. So arguably we need a "formal" definition for the limit in this case too: an analogue of $\varepsilon$-$\delta$ for $\mathbb{C}$.
Limits:
Consider the claim
$$\lim_{z \to \zeta} f(z) = L$$
for $z,c,L$ possibly complex. We define as meaning
$$(\forall \varepsilon > 0)(\exists \delta > 0) \Big( 0 < |z-\zeta| < \delta \implies |f(z) - L| < \varepsilon \Big)$$
This identically mirrors the case for real-valued, real-variable functions. Bear in mind $|\cdot|$ denotes the complex-number magnitude or norm in this case.
Brief Notes on Integrals:
As one might expect, the usual definition of the Riemann integral (in terms of Riemann sums) holds in the way one would expect as well for functions $f : \mathbb{R} \to \mathbb{C}$, in light of these considerations of complex limits.
Contour integrals can also be introduced, if we use functions $f : \mathbb{C} \to \mathbb{C}$. If we want to integrate $f$ over a complex contour $C$ which parameterized by $z(t)$ for $z : [a,b] \to \mathbb{R}$, then we may define
$$\int_C f(z) \, \mathrm{d} z := \int_a^b f(z(t)) z'(t) \, \mathrm{d} t$$
One may also define the integral of $f : \mathbb{C} \to \mathbb{C}$ over $E$ that satisfies $f = f_r + i f_i$ (for $f_r,f_i$ ordinary, real functions) by the rule
$$\int_E f := \int_E f_r + i \int_E f_i$$
(as in, this is where you start as the first principle for such functions: this is not an application of linearity, but an outright definition).
Conclusion:
There are some notions, of course, unique to the complex-valued function case. For instance, differentiability of a function $f(x,y) := u(x,y) + i v(x,y)$ requires
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
to be satisfied. (These are known as the Cauchy-Riemann equations.)
On the integration end of things, some noteworthy phenomena that arise are Cauchy's integral formula and the residue theorem, which give ideas on how to handle contour integrals over closed contours, that perhaps contain singularities. They're also really useful in handling certain real-valued integrals with clever manipulations.
Further details on all of these matters in this post and derivations of the properties you seek, though, are best discussed in texts on "complex analysis", which make the study of these more rigorous and formal -- much in the same way "real analysis" rigorizes the study of derivatives and integrals for real-valued, real-variable functions.
