edit: by "things equivalent to" I really mean universal register machines, something that operates on integers and not directly on real numbers.
At most you use them for stuff like P vs NP, because Turing machines are 'only' polynomially slower than register machines, so it doesn't matter.
But when you are talking about linear vs quadratic on something resembling real computers, Turing machines are the wrong model.
And exact arithmetic on Q + i*Q (complex numbers but the coordinates are rational numbers, not real numbers) is perfectly feasible.
Of course, in a real computer, long rational numbers take longer to manipulate than short rational numbers. But it's just a model. Similar to how we often assume we can handle arbitrary integers in unit time, but in reality longer integers also take longer to handle on a real computer (or we just refuse to handle them and stick to u64 only.)
> Why? It's just a model.
Either a model can be wrong or it is just a model. If it's the former, it matters whether Turing machines are less wrong than exact complex arithmetic machines.
There is, of course, the very rich field of Numerical Analysis which deals with how we actually represent these things in real life, but for a starting point assuming this sort of thing is par for the course.