I doubt there is such a co-hyponym (if we can call it like that) to horizontal and vertical. You'll need to use an alternative name.
If you imagine the 3 axes, then the Z would appear "on the same level" as the X one. Depending on which ones you consider, 2 of the 3 will appear as such and actually, they are.
If you look at the Wikipedia page for Cartesian Coordinate System, under the section Cartesian Space it says:
For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate.
Emphasis mine. It says they are rarely used, but I doubt there are many other alternative terms, other than Z-axis, depth, and so on; they're the most appropriate terms, if you're looking for something technical.