The function $z\mapsto|z|^2$ is not the typical "complex function" that aspires to be analytic, because it is real-valued to begin with. The latter fact makes it possible to talk about contour-lines, while a truly complex function $f:\ {\mathbb C}\to{\mathbb C}$ has no contour lines: The solutions to an equation of the form $f(z)=w_0\in{\mathbb C}$ typically form a set of isolated points in the $z$-plane.
Any "complex function" $f:\quad {\mathbb C}\to{\mathbb C}, \qquad z\mapsto w:=f(z)$ can be viewed as a vector-valued function ${\bf f}:\quad{\mathbb R}^2\to{\mathbb R}^2\ , \qquad{\bf z}\mapsto{\bf w}={\bf f}({\bf z})$ resp. as a pair of functions $(x,y)\ \mapsto \bigl(u(x,y),v(x,y)\bigr)$ via the identifications ${\bf z}:=(x,y)=x+iy=:z$, and similarly for ${\bf w}$. The Jacobian $J_{\bf f}({\bf z}_0) =\left[\matrix{u_x(x_0,y_0) & u_y(x_0,y_0) \cr v_x(x_0,y_0) & v_y(x_0,y_0) \cr}\right]$ of such an ${\bf f}$ at a given point ${\bf z}_0=(x_0,y_0)$ can be any $(2\times2)$-matrix and describes a certain linear map from the tangent space at ${\bf z}_0$ to the tangent space at ${\bf w}_0={\bf f}({\bf z}_0)$.
When such a function $f$ resp. ${\bf f}$ is analytic then the Jacobian of ${\bf f}$ at a point ${\bf z}_0$ can no longer be an arbitrary matrix. The fact that one has an approximation of the sort $f(z_0+h)-f(z_0)= C\ h + o(|h|)\qquad (h\to 0\in{\mathbb C})$ for some complex factor C=:f'(z_0)\in{\mathbb C} implies that $J_{\bf f}({\bf z}_0)$ is a matrix of the form $\left[\matrix{A&-B\cr B & A\cr}\right]\ .$ Geometrically this means that {\bf f}'({\bf z}_0) is a (proper) similarity with stretching factor $\sqrt{A^2+B^2}$ and turning angle $\phi:=\arg(A,B)$. The $A$ and $B$ appearing in this matrix are related to f'(z_0) via f'(z_0)=A+iB.
For an analytic function $f$ these facts must be true not only at a single point $z_0$ in the domain of $f$ but for all points $z_0$ in the domain of $f$. This is expressed in the so-called Cauchy-Riemann differential equations $u_x=v_y$, $u_y=-v_x$.