Is there a third dimension of numbers? Is there a third dimension of numbers like real numbers, imaginary numbers, [blank] numbers?
Alas, there are no algebraically coherent "triplexes". The next step in the construction as has been said already are "quaternions" with 4 dimensions.

Many young aspiring mathematicians have tried to find them since Hamilton in the 19th century. This impossibility links geometric dimensionality, fundamental properties of polynomial equations, algebraic systems and many other aspects of mathematics. It is really worth studying.

A quite recent book by modern mathematicians which details all this for advanced college undergraduates is Numbers by Ebbinghaus, Hermes, Hirzebruch, Koecher, Mainzer, Neukirch, Prestel, Remmert, and Ewing.

However, the set of quaternions with zero real part is an interesting system of dimension 3 with very interesting properties, linked to the composition of rotations in space.
There are of course, examples of R3, in the form X, Y, Z, which behave as three separate real numbers, with an invarient when X -> Y -> Z -> X applies.  

Such numbers turn up in the study of the heptagon and enneagon.  The integer systems are discrete points forming a lattice in this space, the invariants cycle through the solutions to the heptagonal ($x^3-x^2+2x-1=0$) and enneagonal equations, which transforms eg {7} to {7/3} to {7/2}.  

Such a projection converts an infinitely dense arrangemet of points onto a sparse lattice in three dimensions.  That is, in $x,y,z$ form, there is a sphere of size where there are no more than one legitimate value.  So if you evaluate a value even approximately in the three coordinates, it is possible to find the exact value.

One can, if one considers the span of numbers like $1.801937736$, $-1.2469796037$ and $0.445041867912$, which serve for the heptagon, the same role that $1.618033$ and $-0.618033$ do for the pentagon, then the cyclic rotation of this list will preserve multiplication relations, etc.  eg suppose that the order is $p, q, r$  Now $p^2=2-q$ and $q^2=2-r$, and $r^2=2-p$.  One notes that $p+q+r=p*q*r=1$.  These $p$, $q$, $r$ are the solutions to the cubic $x^3-x^2-2x+1=0$ in much the same way that 1.618 and -0,618 solve $x^2-x-1=0$.

The product, for example of $ap+bq+cr$ and its transforms ($aq+br+cp$ and $ar+bp+cq$), is always an integer, speciffically the product of any of these: 7, cubes, and of primes of the form $7n+1$, $7n-1$.

Since one can suppose that there is a real hypercomplex plane, of the form of numbers $a+bj$, where $j=\sqrt{1}$. The conjucate is $a-jb$. The pentagonal numbers belong here.  It features unit hyperbolae, and two 'real' axies, where the values which the value and its conjucates project onto.

And if one can do it meaningfully with two conjucates, it's possible with three, four, five, &amp;c.  I use such systems extensively when wrangling with polygons.