Under the geometric (clifford) algebra of the 2d real plane, complex differentiability is no more restrictive than differentiability on the real 2d plane, and both can be handled with the same formalism.
The geometric algebra of the 2d plane has not just scalars and vectors but bivectors, which are "directed areas". Since a 2d plane has only one area, the space of bivectors is essentially a zero-dimensional vector space. Linear combinations of scalars and bivectors behave like complex numbers.
Differentiability under geometric calculus is the condition that $\nabla A = 0$ for any multivector field $A$. This includes ordinary vector fields as well as "complex-valued" fields. Here is a simple illustration. Let $w(r) = u(r) + e^x e^y v(r)$ be the analogue of a complex field on the real 2d plane with basis $e_x, e_y$. Then,
$$\nabla w = (e^x \partial_x + e^y \partial_y) (u + e^x e^y v) = e^x (\partial_x u - \partial_y v) + e^y (\partial_y u + \partial_x v) = 0$$
These are the Cauchy-Riemann conditions, but our algebra does not actually have complex numbers. This is handled entirely with objects constructed from the 2d real plane.
Now consider a (co)vector field $f = f_x e^x + f_y e^y$. The differentiability condition reduces to
$$\nabla f = (\partial_x f_x + \partial_y f_y) + e^x e^y (\partial_x f_y - \partial_y f_x) = 0$$
The two terms produce the condition that the vector field have vanishing divergence and curl.
How are these concepts related to complex differentiability and integrability? Through this formalism, we can see that a lot of the concepts in complex analysis carry over to vector calculus. When $\nabla f = 0$ in this way, we can find the value of $f$ at any given point like so. Let $\nabla G = \delta(r)$ be the Green's function for the vector derivative. Then,
$$\oint G(r-r') \, d\ell' \, f(r') = -\int \delta(r-r') \, dA' \, f(r') = -i f(r)$$
Look at the left-hand side: this is an integral over a curve. Look at the right: this is proportional to the value of the function at a point. The value of a function is determined entirely by its values on a bounding curve? That's a result from complex analysis! But this is a theorem in vector calculus! In fact, they're both just aspects of the same bigger idea, the generalized Stokes theorem, or the fundamental theorem of calculus.
The calculus of complex fields is just a subset of the calculus of multivector fields on the 2d plane, and as such, so many of complex analysis' results apply to vector fields also--the neat conditions on integrability and so on. That complex analysis is the dominant way of looking at the 2d plane is, to me, quite lazy and misses how vector fields with the right properties can be neatly tied into those theorems too.