The field with one element $\mathbb{F}_1$ (a.k.a. F-un).

Having a precise notion of such a field would allow us to further exploit the analogy between number fields and function fields (much like discrete valuation rings, Dedekind domains, schemes, etc. have done in the past). In particular, with a suitable notion of $\mathbb{F}_1$ we should be able to find a proof of the Riemann hypothesis based on Weil’s proof of the Riemann hypothesis for curves over finite fields.

The integers are not an algebra over any field in the classical sense, which makes it a priori impossible to adapt Weil's argument to this case. For this analogy to work, a good definition of $\mathbb{F}_1$ should have the property that $\mathbb{Z}$ is an $\mathbb{F}_1$-algebra.

Furthermore, there are lots of connections with “q-deformations”. I don't know much about this, but John Baez has some nice stuff written at This Week's Finds in Mathematical Physics, weeks 183-187.

An example of the cool unification we can do with $\mathbb{F}_1$ at our disposal: we can make sense of statements like

Combinatorics is linear algebra over $\mathbb{F}_1$.

(This idea is originally due to Jacques Tits, if I recall correctly.)

For a possible way of rigorously developing such a theory, see Nikolai Durov's paper New Approach to Arakelov Geometry.