You could try the following axiomatic definition of the complex numbers that I just made up.
Definition. A quadruple $(K,+,\cdot,i)$ is a complex field if the following axioms are satisfied:
- $(K,+,\cdot)$ is a field containing $\mathbb R$ as a subfield.
- $i\in K$ and $i^2=-1$.
- For all $z\in K$, there exists $a,b\in\mathbb R$ such that $z=a+bi$.
(For simplicity, I will yield to the usual abuse of notation whereby an algebraic structure is referred by the name of its underlying set.)
We can easily deduce a few basic properties of complex fields from these axioms: for instance, if $a+bi=0$ with $a,b\in\mathbb R$, then $(a+bi)(a-bi)=0$, hence $a^2+b^2=0$ and so $a=b=0$. It follows that if $a+bi=c+di$, then $a=c$ and $b=d$. Continuing in this manner, we can deduce all of the basic results about complex fields, and then move onto proving results in complex analysis. This is in the same way that all results in real analysis can be deduced on the basis of the fact that $\mathbb R$ is a complete ordered field – at no point does the precise construction of $\mathbb R$ play any role. The constructions of $\mathbb R$ and $\mathbb C$ respectively are arguably only important to the extent that they tell us that complete ordered fields and complex fields exist – without them, there is a danger that everything we deduce is in fact vacuous. On the other hand, once we have carried out the respective existence proofs, and shown that $\mathbb R$ and $\mathbb C$ are unique up to unique isomorphism, we can mentally discard their set-theoretic constructions. It is best to think of $\mathbb R$ as denoting an arbitrary (but fixed) complete ordered field, rather than a collection of Dedekind cuts, say. The same goes for $\mathbb C$.
The proof that there is a unique up to unique isomorphism complete ordered field is well-known, and is covered in most textbooks in real analysis, say Spivak's Calculus. Here I would like to prove that complex fields are similarly unique. There is a natural notion of "complex field homomorphism" between complex fields $K$ and $K'$: it is a field homomorphism $\varphi:K\to K'$ which fixes $\mathbb R$ in place and sends $i_K$ to $i_{K'}$. If $K$ and $K'$ are complex fields, and $\varphi:K\to K'$ is a homomorphism, then for all $a,b\in\mathbb R$,
$$
\varphi(a+bi_K)=\varphi(a)+\varphi(b)\varphi(i_K)=a+bi_{K'} \, .
$$
It follows that there is just one homomorphism between $K$ and $K'$, and it is an isomorphism sending $a+bi_K$ to $a+bi_{K'}$. Thus, there is no real harm in speaking of the complex field. Note that to ensure the isomorphism is unique, we had to specify a choice of square root of $-1$, and insist that complex field homomorphisms leave every real number alone.
From the ordered pair construction of $\mathbb C$, we know that complex fields exist*, and the above paragraph shows that they are unique in a suitable sense. Hence, we can indeed do complex analysis simply on the basis of the axioms of complex fields mentioned above.
*There is a slight technical issue with the ordered pair construction of $\mathbb C$: if we define $\mathbb C$ as $\mathbb R^2$, then $\mathbb R$ is not literally a subfield of $\mathbb C$, and hence $\mathbb C$ is not a complex field. (Similarly, $\mathbb R$ is not literally a subfield of $\mathbb R[x]/\langle x^2+1\rangle$.)
This issue can be patched in a number of ways. For instance, we could take $\mathbb C$ to be $\bigl(\mathbb R^2\setminus \{(a,0)\mid a\in\mathbb R\}\bigr)\cup\mathbb R$ instead. Alternatively, we could modify axiom (1) by simply requiring that there is an embedding $\xi$ of $\mathbb R$ in $\mathbb C$. To ensure that complex fields are unique up to unique isomorphism, we would then have to insist that the embedding of $\mathbb R$ in $\mathbb C$ is part of the structure of a complex field, and that homomorphisms of complex fields respect the emeddings.