In physics, time and space are considered fundamental and can not be defined in other terms. Periodic events and periodic motion, like the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, heartbeats, have long served as standards for units of time.
Spacetime is a notion that puts time on the same footing as the visible three spatial dimensions. But if extra spatial dimensions can be added, in some theories, to help assimilate all physical forces into a unified model of reality, can there be another dimension of time, as well?
Now, the dimension of time has an important role in describing matter, gravity and other forces of nature, but something doesn't fit. It's like looking at a projection of a three-dimensional object onto a two-dimensional wall, that can have many different shapes, but no individual shape can fully display all the properties of the object.
Maybe a two-time description of dynamics could explain some properties of dynamical systems that are yet to be explained. It seems that by adding an extra time dimension, along with a requirement that all motion in the enlarged space be symmetric under an interchange of position and momentum at any instant, can apparently explain all the possible dynamics in ordinary spacetime.
In fact, alternative time dimensions could already exist, curled up in little balls, so tiny to notice that if you moved through one of those dimensions, you'd get back to where you started so fast that you'd never realize that you had moved.
The existence of another time dimension could bring a general unification of dynamics, but it could also explain other phenomena, like the CP conservation - C for charge conjugation, which transforms a particle into its antiparticle, and P for parity, which creates the mirror image of a physical system - in the strong interactions described by QCD without the need for a new particle, the axion, which has not been found in experiments.
It could also explain the fact that the elliptical orbit of planets remains fixed (not counting well-known tiny precessions) in what is known as the "Runge-Lenz" symmetry effect, that remained somewhat mysterious in the study of celestial mechanics and now could be understood as being due to the symmetry of rotations into the fourth space dimension.