Kronecker. Nuff said. Even the numbers themselves historically started as positive integers and were subsequently generalized to hell and back. Here are some other well known concepts that "should" involve $\mathbb{N}$ but were generalized to $\mathbb{Q}$, $\mathbb{R}$ or even $\mathbb{C}$:

- Dimension $\rightarrow$ Hausdorff dimension.
- Factorial $\rightarrow$ gamma function.
- Differentation $\rightarrow$ half-differentation (etc.)

So, can you extend this small to a big list?

(Motivation: Some hypothetic knot polynomial I calculated with demanded a dimension of its associated group representation - thus the "rt" tag - of 60/11. That is noooooot boding well for its existence. :-)