The best way to explain the difference, as I get it, is that the meaning of a derivative in complex analysis is much, much more restrictive than that in real analysis. This gives complex-differentiable functions much more structure than their counterparts in real analysis.
This can be thought to have something to do with having more dimensions. In particular, 2D geometry has more structure than 1D. To understand just how they are "much more restrictive" geometrically, we can think of a differentiable real function to be one that looks, on a small scale, like a "stretch" of the line around that area. But a complex-differentiable function on the plane has to actually look like a shape-preserving transformation in a small area. It cannot stretch (irregularly), reflect, or otherwise change shapes. It can only change scale or rotate. Much more information is required to be preserved, and this makes the functions more restricted and mathematically more interesting. In general, the more rules your widgets have to follow, the more interesting they tend to be.
More formally, a general 2D differentiable function from the 2D plane to the 2D plane is one where its partial derivatives all exist everywhere, so we have a Jacobian matrix. To be complex-differentiable, we essentially add the requirement that this matrix looks like a matrix representation of a complex number. (The equations on the partial derivatives that this gives are the Cauchy-Riemann equations.) This complex number is the function's complex derivative.
Sequences in the complex numbers aren't much different from real numbers: they can be thought of as just pairs of real numbers. In particular, there is nothing magic about convergence, limits, etc. of sequences of elements of $\mathbb{C}$ versus those of elements of $\mathbb{R}^2$. In this sense there isn't as much "new" to get in sequence stuff. The new stuff really comes from differentiable functions and the much more restrictive definition of complex derivative. The thing that sets the complex numbers apart from just plain old pairs of real numbers is their multiplication operation, which entangles the real and imaginary components together. And it is in the derivative where this operation is exploited, by the division by the complex increment $\Delta z$. In addition, a limit of a complex function, which is what the difference quotient is, as opposed to the limit of a sequence, is also more restrictive, since we have far more "angles of attack" we have to account for in 2 dimensions instead of in 1 dimension.
By definition $\operatorname{sgn}(z) = z/|z|$ is a unit vector. As such, $|z| = z/\operatorname{sgn}(z) = \overline{\operatorname{sgn}(z)} * z$.
This is because for any unit vector $u \in \mathbb{C}$, we have $1/u = \overline{u}$.
Best Answer
Let $z \in \def\C{\mathbf C}\C$ be any complex number, then $\def\Re{\operatorname{Re}}\def\Im{\operatorname{Im}}$ $$ z = \Re z + i\Im z $$ multiplying by $i$, we have $$ iz = -\Im z + i\Re z$$ that is the real part of $iz$ is $\Re(iz) = -\Im z$, so we get $$ \Im z = -\Re (iz) $$ Folland applies this to $z = f(x)$, giving $$ \Im f(x) = -\Re [if(x)] $$ As $f$ is complex linear, $if(x) = f(ix)$, giving $$ \Im f(x) = -\Re f(ix) $$ As $u$ was defined to be $\Re f$, we have $$ \Im f(x) = -u(ix) $$ Hence \begin{align*} f(x) &= \Re f(x) + i\Im f(x)\\ &= u(x) + i \cdot \bigl(-u(ix)\bigr)\\ &= u(x) -iu(ix) \end{align*}