First, I comment on the idea of 'construction.' When analysis books today refer to the construction of the real numbers (or rationals, etc), they usually refer to some sort of explicit extension of something number of axioms. Many people have no trouble 'believing' in the natural numbers, and so start there. Groups are lovely structures (a set with an associative operation, an identity, and inverses) that are well-studied, and so one might add zero and the negative numbers to get a group. But fields (groups with two operations, sort of, that commute and distribute) are great structures too, and there are ways to turn the familiar system of rationals numbers into a much bigger set - the real numbers. The general ideas of this construction often falls under the idea of making the rationals 'complete' - making it so that any Cauchy sequence of rationals (a sequence of rationals such that the differences between terms becomes arbitrarily small, sort of) converges. And then one can sort of complete this process with the complex numbers.
I say 'complete this process' because all of our standard mathematical operators are closed in the complex numbers. All polynomials completely factor, we can take roots and multiples, multiply by inverses, etc. without fear of getting something meaningless. And the constructions in analysis books usually go in specific ways that amount to assuming the least amount of things to get the greatest amount of information. But I want to point out that this isn't really how the systems themselves got started (except perhaps the complex numbers - they have their own history). These different systems had a much less clear progression. It is not as if one day, Pythagoras woke up and said, "Hmm. I wish that my triangles lengths would exist. So I will extend my normal ratios into something beautiful." Instead, Greeks panicked when they thought there were things so imperfect as an irrational number.
But one aspect is preserved from the original development of these number systems: the idea that some systems aren't closed under some sort of operation, and so we try to expand it. First came the addition of 0 (absent for much of history). Ratios are very old, and strangely enough are often older than the concept of a zero in various areas of the world. They're just so... useful. But then the Greeks found irrational numbers (Pythagoras, for instance, suggested that there was no common measure between the hypotenuse and leg of a 45-45-90 right triangle - i.e. it was irrational). This was a very big idea, and this took a very long time to sink in. Then things like Cardano's solution to the cubic used imaginary numbers in meaningful ways - whoa. Taking square roots of negative numbers is a new trick. And the funny thing is that he didn't even use his formula to produce imaginary numbers - they just happened to appear and then later cancel in the middle steps. Another big idea. But by then, our current formal style was starting to get around. Not too long afterwards, the idea of fields and field extensions came around, and so we began to consider what things were and weren't closed.
The basic idea of a field extension is that we take some field (like the rationals) and we take something not in that field (perhaps the root of a polynomial - the standard idea). And then we see how much larger we need to make that field to accommodate that extra bit. Sometimes, it's small (extending the rationals to hold the root of $x^2 - 2 = 0$, for instance, is a very small extension). Sometimes, it's big (to hold something like $\pi$, for instance). Now, the big idea is that no root of a polynomial with complex coefficients lies outside of the complex numbers. So one can't extend the complex numbers in that fashion. And in that sense, the complex numbers are as 'big' as it gets.
But one might ask whether or not it is possible to meaningfully go beyond the complex numbers. And the answer is... sort of. There is something called the Cayley-Dickson Construction that makes arbitrarily 'bigger' algebras than the complex numbers. But the problem is that all the nice algebraic properties quickly break down. The quaternions, the one 'after' the complex numbers, don't even commute!
And there are perhaps less-extreme extensions, but of a different calibre. For instance, one might do things called 'compactifications' that usually sort of amount to pretending that infinity was actually a number of sorts in our space. This is something from the realm of topology, so it's a bit more abstract. But one might give rise to the analysis of projective spaces or the Riemann Sphere in this way. But the 'operations' that these facilitate are not really numeric in nature, and so this is a fine line.
So in short - the complex numbers are adequate in almost every way we could hope for. But there are things that are bigger in some sense or another.