One way to sharpen the question is to stop asking whether C is "fundamental" and instead ask whether it is forced by mild structural constraints. From that angle, its status looks closer to inevitability than convenience.
Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior. You essentially land in C, up to isomorphism. This is not an accident, but a consequence of how algebraic closure, local analyticity, and linearization interact. Attempts to remain over R tend to externalize the complexity rather than eliminate it, for example by passing to real Jordan forms, doubling dimensions, or encoding rotations as special cases rather than generic elements.
More telling is the rigidity of holomorphicity. The Cauchy-Riemann equations are not a decorative constraint; they encode the compatibility between the algebra structure and the underlying real geometry. The result is that analyticity becomes a global condition rather than a local one, with consequences like identity theorems and strong maximum principles that have no honest analogue over R.
I’m also skeptical of treating the reals as categorically more natural. R is already a completion, already non-algebraic, already defined via exclusion of infinitesimals. In practice, many constructions over R that are taken to be primitive become functorial or even canonical only after base change to C.
So while one can certainly regard C as a technical device, it behaves like a fixed point: impose enough regularity, closure, and stability requirements, and the theory reconstructs it whether you intend to or not. That does not make it metaphysically fundamental, but it does make it mathematically hard to avoid without paying a real structural cost.
This is the way I think. C is "nice" because it is constructed to satisfy so many "nice" structural properties simultaneously; that's what makes it special. This gives rise to "nice" consequences that are physically convenient across a variety of applications.
I work in applied probability, so I'm forced to use many different tools depending on the application. My colleagues and I would consider ourselves lucky if what we're doing allows for an application of some properties of C, as the maths will tend to fall out so beautifully.
Not meaning to derail an interesting conversation, but I'm curious about your description of your work as "applied probability". Can you say any more about what that involves?
Absolutely, thanks for asking!
Pure probability focuses on developing fundamental tools to work with random elements. It's applied in the sense that it usually draws upon techniques found in other traditionally pure mathematical areas, but is less applied than "applied probability", which is the development and analysis of probabilistic models, typically for real-world phenomena. It's a bit like statistics, but with more focus on the consequences of modelling assumptions rather than relying on data (although allowing for data fitting is becoming important, so I'm not sure how useful this distinction is anymore).
At the moment, using probabilistic techniques to investigate the operation of stochastic optimisers and other random elements in the training and deployment of neural networks is pretty popular, and that gets funding. But business as usual is typically looking at ecological models involving the interaction of many species, epidemiological models investigating the spread of disease, social network models, climate models, telecommunication and financial models, etc. Branching processes, Markov models, stochastic differential equations, point processes, random matrices, random graph networks; these are all the common objects used. Actually figuring out their behaviour can require all kinds of assorted techniques though, you get to pull from just about anything in mathematics to "get the job done".