What you have done is embed the complex plane into three dimensional space. You are certainly free to do this, but the space you get won't be as natural or have the same beautiful properties as the complex plane. So maybe the key to answering your question is explaining why the complex plane is so natural and beautiful.

To begin, the complex plane is an example of a what is known in algebra as a field. A field is a space where that usual operations of algebra - addition, subtraction, multiplication, and division - are possible and satisfy a certain natural list of properties. The real numbers also form a field, and the field of real numbers sits inside the field of complex numbers in such a way that the complex numbers form a two dimensional space over the real numbers. This means that the complex numbers form what is called a "division algebra" over the real numbers. There is another division algebra over the reals: it is the space of so-called quaternions, which contains 3 essentially different square roots of -1 and thus has dimension 4 over the real numbers. However, this space isn't quite a field because multiplication doesn't commute; a*b is not always the same as b*a.

My point in explaining all of this is that there is a theorem, called the Frobenius theorem, which says that there are up to isomorphism exactly three finite dimensional division algebras over the reals: the field of real numbers itself, the field of complex numbers, and the division algebra of quaternions. So this is the first thing that is very natural about the complex numbers which is not shared by your construction: because of this theorem, I can tell you that no matter how hard you try you will never be able to define a nice notion of multiplication and division in your space which make it into a field or even a division algebra.

There is another reason why the bare complex plane is the right object to consider without any embellishment. The real numbers form a very nice space, but they have a crucial defect from the point of view of algebra. Specifically, there are lots of polynomials with real coefficients that do not have real roots; x^4 + 5x^2 + 2, for example. The simplest example of such a polynomial is simply x^2 + 1, whose root would have to be a square root of -1. It is well known that the complex numbers are formed from the real numbers by adjoining a root, typically called i, of this polynomial. The fact which is not as well known (and whose proof is even lesser known) is that making sure this one little polynomial has a root guarantees that every other possible polynomial also has a root, including x^4 + 5x^2 + 2 as well as sqr(52)x^9 - e^3 x^4 + log(pi) x - 87.2. Even if we allow polynomials with complex coefficients, they still have roots; it is not a priori obvious that just because -1 has a complex square root, i must also have a square root and so must the square root of i. This property of the complex numbers - that every polynomial with complex coefficients has complex roots - means that the field of complex numbers is "algebraically closed". And it is a fact that the complex numbers are the smallest algebraically closed field containing the real numbers as a subfield. So even if we could enlarge them, why would we want to?

My final comment about the complex numbers involves a bit of calculus. Calculus over the real numbers can be a bit irritating at times, because so many weird things can happen. We can have differentiable functions whose derivatives are not continuous, such as f(x) = x sin(1/x), and hence can't be differentiated again. Likewise, we can find 5 times differentiable functions which are not 6 times differentiable, and so forth. If we are lucky enough to find a function that can be differentiated infinitely many times, we would like to use the theory of Taylor series to calculate that function using power series, but even this is not always possible: f(x) = e^(-1/x) is infinitely differentiable at 0, but its derivatives are all 0 there and so f is a nonzero function whose Taylor series is 0. Not good.

One of the most beautiful thing about the complex numbers is that almost all of these difficulties magically disappear. Continuous functions on the complex plane are not necessarily differentiable, but once you know a function is differentiable near a point it is automatically infinitely differentiable and, even better, its Taylor series converges to it in some neighborhood of the point. This is really just the tip of the iceberg when it comes to all of the completely amazing properties that differentiable functions of a complex variable have, but it is not a property that functions on your space will possess. Such functions will be very nice when they are restricted to a two dimensional component of your space (as long is it contains the imaginary axis and is viewed as the complex plane), but all of the usual house of horrors can manifest themselves along the third axis.