Physicist: This is a really nasty, complicated question. It’s isn’t remotely straight-forward in the way that adding spacial dimensions is. The universe we live in is “3+1 dimensional”, meaning 3 spacial dimensions and one temporal dimension. While time and space do have more in common than you might think, they are still (no surprise) fundamentally different.
Because we live in a 3-D space, when we talk about what life is like in higher (spacial) dimensions we’re free to extrapolate from what we know life would be like in one dimension, how that changes when you move up to two dimensions, and how things are further generalized in three dimensions. We can expect those patterns to continue into higher dimensions (example here!). We’re even free to laugh haughtily at the pitiable denizens of a hypothetical one-dimensional world who are incapable of seeing how things behave in 2 or 3 dimensions.
Well, in terms of time that’s exactly the situation we’re in. In the same way that a one-dimensional critter can know everything about where they are with a single number (like points along a ruler), a one time-dimensional critter (for example, everyone and everything) can know everything about when it is with a single number. The fact that we use several numbers to designate time is an indulgence.
Going from talking about points on a line (1-D) to talking about points in a plane (2-D) is a huge leap. Suddenly have to concern yourself with trigonometry. In a 3+2 dimensional universe, “temporal cartographer” could be a real job, and the working day would be “9 to 5 by 9 to 5″.
I wish I could offer up a reasonable guess about what life would be like with multiple time dimensions. But just as a 1-D person can’t conceive of turning around, I can’t say what it means to “turn corners in time”. Normally when presented with “what if” questions, you can ponder it in terms of how the laws of physics would change the least. In this case they’re entirely up in the air. For example, in physics it’s sometimes important to show that the past and future are different “places”. This is so “obvious” that we take it for granted. Regular rotations give us a way of “translating” any point into any other point that’s the same distance away. That is; if you’re looking at something and you turn around, then that thing is behind you (physics is full of profound truisms like that). Special relativity has provided us with another kind of “rotation” that exchanges some of one of the space directions with some of the time direction, but in a not-quite-as-simple way that involves a new kind of distance.
In regular space you can rotate, and in so doing, the relative position of everything around you traces out a circle. In particular, things in front of you can end up behind you (Try it! This post can wait.). Rotation is just an interchanging of two space directions. With special relativity comes the idea of the “Lorentz boost”, which is just a fancy way of saying “space-and-time rotation”. When you go from sitting still to riding on a train, you’ve undertaken a Boost. In the same way that physically turning around rearranges where things are (with respect to you), a Boost rearranges where and when events take place (with respect to you). For example, when you’re not riding the train it shows up in lots of places at different times, but when you do ride it, it only shows up in one place at different times. However, and this is the important part, the Lorentz Boost can’t take an event that’s in your future and rotate it into your past, or vice versa.
However! With another time direction comes a new kind of rotation. Ordinary rotation is an interchange of two space directions, Boosts are an interchange of a space direction and the time direction, and in 3+2 D space you can have a rotation that exchanges the two time directions. Importantly, this new rotation can smoothly take events in your future and take them into your past. That is to say; in 3+2 dimensional space you should be able to “turn around in time” and face the past.
I have no idea what that means.
It may be the case that some of our physical laws are symptoms of the dimension we live in. For example, in a 1+1 D universe you’d have “conservation of directionality”, because nothing can turn around. In our 3+1 D universe the fact that particles are only Fermions or Boson can be tracked back to the fact that we live in more than two (spacial) dimensions (very complicated details here). However, there may be a lot of “laws” that are caused, at least in part, by the restrictions placed on us by the single time dimension we have to work with.
So, unfortunately, there are no actual answers to what the world would be like with more time dimensions, but (since it has nothing to do with reality) there’s no hurry to find those answers.
Answer gravy: Many of the most basic laws involve equations that are “ill-posed” in multiple time dimensions, and either don’t have solutions, or don’t make sense. Almost every law in physics is written in terms of cause and effect, initial conditions to later conditions. Extending that doesn’t sound terrible. It seems like you could just extend the laws we have now, the same way you can for spacial dimensions, to work the same on each time direction the same way it does for just one. But the laws we work with in physics just don’t extend in any useful way into higher dimensions without tacking on lots of weird extra restrictions that, in all but name, bring you back to a 3+1 dimensional universe.
The result of the initial conditions from one time direction will usually disagree with the initial conditions on the other times. No matter how you define the initial conditions (what “initial” means in multiple time dimensions is an issue in itself), you’ll find that the initial conditions always cut across “characteristic lines”, which (this is not obvious) lead to a lack of solutions in general. “Characteristic lines” are the paths that solutions to an equation “propagate along”, and having initial conditions on them basically means putting more information onto what should already be a solved problem.
For example, the sound that a speaker generates can be described easily using basic acoustic laws: the sound created by the speaker (initial condition) leads naturally to an easily calculable sound everywhere else (final conditions). However, the sound created by a speaker traveling at the speed of sound, cannot be easily calculated, because the sounds the speaker makes all “stack up”; the speaker is continuing to make new sound on characteristic lines. There are ways to deal with this, but they’re not “basic”, and required a lot more research. The same mathematical complications crop up in effectively everything when multiple time dimensions are considered.
There’s a paper here that considers some workarounds in detail.