I saw, and experienced, this approach in school and see no value in it for improving your understanding complex numbers. The point of the approach is to teach students about abstraction and formalism, and complex numbers happen to be a convenient place to do it. But the formalism is a barrier for building up a more convenient mental model about what is really going on.
Before disputing this, in any calculation that you've ever done by hand with complex numbers, do you naturally write it as (a, b) or a + bi? I always do the latter, and it saves me both time and conceptual effort.
And a random note. If you go on past advanced Calculus, you'll encounter two subjects that take Calculus and go back to the basic foundations and build them up. The first is real analysis, for which you have to learn all of the ways that things fail to work out like you would want them to. The other is complex analysis, where you wind up learning all of the ways that everything has to work out amazingly perfectly.
The difference between the two subjects is that "differentiable" in the 2-dimensional structure of complex numbers is a far, far stronger condition than "differentiable" is for the real numbers. Indeed there actually exist functions that you can construct which are infinitely differentiable everywhere in the real numbers, but for which on no interval can you extend them to a function that is differentiable in the complex plane.
> But the formalism is a barrier for building up a more convenient mental model about what is really going on.
Actually, I'd say that complex numbers are a wonderful opportunity for illustrating that math is nothing but a mental model, not necessarily with anything "really going on", AND at the same time incredibly useful.
IMO the best way to think about complex numbers is to start with the observation that sqrt(-1) cannot exist in the realm of real numbers, quite easily provable. Then you make the bold-assed assumption that it exists anyway and simply call it i. Then play around with it a little, and find out that adding the axiom "there is something called i, where i * i == -1" does not lead to any contradictions; instead you can work with this i just fine, and numbers of the form a+bi are closed under all your everyday operations, just by applying your everyday arithmetic rules. Neat!
Then you notice that those numbers can be interpreted as 2D vectors and some operations on them as geometrical transformations thereof. Super Neat!
Then someone notices that those numbers and the operations on them can actually be used to model certain aspects of the physical world. Holy Crap!
First, I find that having more different ways to look at a problem, the better I am able to deal with it. I can look at an equation algebraically, or as a graph, e.g. I can use rectangular coordinates or polar coordinates. I can look at complex numbers as abstract entities or as points in a plane.
Second, if you look at the history of complex numbers, mathematicians were just not sure what to make of them, and had no way to have confidence that what they were doing was even consistent. Being able to interpret them as point in a plane with intuitive geometric operations gave them a huge boost.
Third, thinking of them this way led to the search for generalizations. Gauss and Hamilton tried to find a way to do arithmetic in three dimensions, or prove that it couldn't be done. Hamilton eventually found the four-dimensional quaternions. And the (ac-bd,bc+ad) definition was generalized to the Cayley-Dickson construction.
And not long afterwards, Clifford generalized real numbers, complex numbers, and quaternions into what are now known as "Clifford algebras". Handy stuff. Certain algebras allow you to express geometric shapes such as points and lines using very simple equations. Other algebras show how quaternions (for example) arise naturally as the even subalgebra of Cl0,3(R). The "spacetime algebra" appears as CL1,3(R), which makes it easy to express special relativity.
"Young man, in mathematics you don't understand things. You just get used to them."
-- John von Neumann
For what it's worth, I have sometimes manipulated complex numbers as (a, b), usually when using them as 2-D vectors. But I make no claims to being a mathematician.
That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic. It's far easier to come up with (and understand!) examples of functions that are smooth when considered as real maps from R^2 to itself, but nowhere homomorphic when considered as complex functions of a single complex variable, though. Complex conjugation is an obvious example: as the real map (x,y) -> (x,-y), it's linear, so smooth, but only holomorphic at points where 1 = -1, so nowhere. Complex conjugation restricted to the real line does have an obvious holomorphic extension, of course!
That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic.
Did you accidentally get my intended point backwards? My point is that real analysis admits all sorts of fine gradations of counter-examples, and complex analysis does not. In this case I was indeed thinking of a smooth real function that is nowhere analytic. (For non-mathematicians, that would mean a function that can be differentiated any number of times at any point, but which cannot be written as a power series.)
Contrast with complex analysis where continuous at a point and differentiable in a neighborhood of that point implies analytic. Even something as ill-behaved as the point at 0 of the absolute value function is not possible in complex analysis.
Before disputing this, in any calculation that you've ever done by hand with complex numbers, do you naturally write it as (a, b) or a + bi? I always do the latter, and it saves me both time and conceptual effort.
And a random note. If you go on past advanced Calculus, you'll encounter two subjects that take Calculus and go back to the basic foundations and build them up. The first is real analysis, for which you have to learn all of the ways that things fail to work out like you would want them to. The other is complex analysis, where you wind up learning all of the ways that everything has to work out amazingly perfectly.
The difference between the two subjects is that "differentiable" in the 2-dimensional structure of complex numbers is a far, far stronger condition than "differentiable" is for the real numbers. Indeed there actually exist functions that you can construct which are infinitely differentiable everywhere in the real numbers, but for which on no interval can you extend them to a function that is differentiable in the complex plane.