I think the base idea here is something like this: if you want to describe, say, a sound-wave, you can use complex numbers to represent the wave, but you can also, in principle, use strictly real numbers to describe the behavior of each individual molecule of gas using Newton's equations of motion (assuming you can ignore quantum effects for your simulation). So, in classical mechanics, while complex numbers are a useful abstraction for waves, they are not fundamental.
The same is not true for QM: the wave part / the complex numbers are fundamental.
Also note, when people say "complex numbers" they refer to the algebraic field, which is the object composed of [a pair of reals, complex addition, complex multiplication]. In particular, complex multiplication is the key here, since it is what differentiates complex numbers from 2-vectors. To drive this point further, while 1-vectors, 2-vectors, 3-vectors, 4-vectors etc behave very similarly, only the reals and the complex numbers can form a field - there are no "numbers" in the same sense formed of 3-tuples or 4-tuples or other n>2-tuples of reals*.
This is why the discussion focuses on the complex numbers (meaning, again, not just a pair of reals, but also the particular +,-,*,/ operations for them that make them a field).
* to be fair, quaternions come pretty close - you can define a 4-tuple of reals + addition + multiplication that is almost a field, except that multiplication is anti-commutative instead of being commutative (p*q = -q*p).