This is a late night thought, but it seems interesting enough to ask: Has anyone considered the possibility that time might have more directions that forward and backward? Could time go sideways?
I know that there are theories that include multiple dimensions of time (this would be required for a direction other than forward/backward). People have gotten so used to being surprised by counterintuitive truths in physics that a lot of non-typical theories have been developed. They tend to depend on things which are difficult or impossible to verify experimentally (or else they would have been proven incorrect already).
To answer the question directly: I have never seen any proof that time cannot have additional dimensions. There is also a duplicate of this question that comes to the same conclusion here. Therefore, until I come upon such proof, I will assume that time could have additional dimensions.
It's certainly an interesting idea. I might read the wikipedia article now:
Additional sources include a short chat by Neil deGrasse Tyson about this: https://www.youtube.com/watch?v=CvKuEgzElec
A few quick introductory comments: whether time goes in any direction is not universally accepted. Have a look at the answers to Is the flow of time regular? and Is there a proof of existence of time? for some discussion of this question.
As far as relativity is concerned time is just a dimension. We line in a four dimensional spacetime, which means we need four numbers $(t, x, y, z)$ to identify points in spacetime. Time is just one of the four dimensions in spacetime. One way to interpret your question would be to ask if there can be more than one time dimension e.g. could we have a five dimensional spacetime $(t_1, t_2, x, y, z)$ with two time dimensions? This is an idea that crops up from time to time, but the big problem with it is that two time dimensions generally allow closed timelike curves i.e. time travel and that causes havoc with physics because it implies a loss of causality. Since observations suggest the universe does exist and is reasonably predictable, that strongly implies there is only one time dimension.
The physicist Itzhak Bars has developed theories with two time dimensions, and these seem to avoid the worst of the problems though you should note that his approach is not widely accepted. There is also a theory called F-theory that can have two time dimensions, though these aren't really what we would normally think of as time dimensions.