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Are there multiple (small class) extensions of the naturals?

Suppose we have the natural numbers $\mathbb N = \left\{0,\,1,\,2,\,\cdots\right\}$, and the usual addition operator $+$. Suppose also that we want to _extend_ the natural numbers, so we have more numbers to play with. We can define the set $\mathbb Z$ as equivalence classes of pairs of naturals $(a, b)$ and $(c, d)$ under the relation $a+d=b+c$: these are (isomorphic to) the integers. It makes sense to identify the integer 3 with the natural number 3, and so on, treating $\mathbb N \subset \mathbb Z$, in which case $\mathbb Z$ can be thought of as the closure of $\mathbb N$ under additive inverses.

However, if we want to extend this further, we have a choice to make. We can either take the field of fractions of the integers, giving us $\mathbb Q$, or we can choose a prime $p$ and construct the $p$-adic integers. From there, we can construct either the reals ($\mathbb R$), the algebraic numbers, or the $p$-adic numbers. Both the reals and the algebraic numbers can be seen as subsets of the complex numbers ($\mathbb C$), but there are $p$-adic numbers with no complex equivalent, and to my knowledge it is not possible to unify these two branches. There is, therefore, no unambiguous extension of the integers, and it does not necessarily make sense to unify $-2$ with the rational $\frac{-2}{1}$ or with the $3$-adic $\cdots22221$.

This makes me wonder: _can_ we necessarily identify the integer 3 with the natural number 3? Might there be a natural (no pun intended) extension of the natural numbers which does not admit analogues of the integers? Can this notion be firmed up enough to prove the non-existence of such an integer-less extension / generalisation of the natural numbers, under the usual axioms of mathematics?

I know we can consider the natural numbers as equivalent to the finite ordinals, or the finite cardinals (which can be identified with a subset of the ordinals, iff we assume the Axiom of Choice), but the ordinals and cardinals are both proper classes, so there's a sense in which they don't really count. (And there are negative surreals, anyway, so it's only cardinals-without-choice that are suspect.)