What's the purpose of a second dimension when we say that time is two-dimensional? The first dimension refers to the passage of time along the x axis (past and future), but I am wondering what the y axis is for when we postulate that time has two-dimensions. What are some of the philosophical claims made about the possible 2nd dimension in time?
As I see it, this is a physics question.
You might like this video which gives an introduction to two dimensions of time.
Key to thinking about dimensions, is to understand their relation to continuous symmetries, by Noether's theorem. The string model of 10 spatial dimensions & 1 of time has been elaborated here, which can help in an accessible way to picture the role of extra dimensions.
I follow the ideas of Rovelli & others that all dimensions are emergent, from something like a spin network. I would link the holographic principle and associated conservation of information, to understanding us as being on a 4D surface in 5D, with a fractional fractal dimension, as the surface expands in a regular way through quantum uncertainty (into Many Worlds). This can account for the arrow of time as an emergent directional symmetry.
Time is different from space in having a directional symmetry. An additional time dimension would have to also have such an assymetry, directionality, to make events in relation to it irreversible.
More information is needed here to determine what it is you wish.
To begin with, who asserts that time is two-dimensional? To a physicist, two-dimensional time represents a mathematical formalism that is useful for transforming certain physical phenomena between mathematical regimes, to make their interpretation more clear. On the other hand, to a philosopher, two-dimensional time means anything (s)he wants it to mean.
Having two dimensions of time in the physical realm would mean that in addition to being able to go forward and backward in time, you could also move sideways in time, a concept which would abolish causality as follows: from one's current position in space and time, one could move sideways in time onto a new world line, then forward or backward, and then sideways again to join up again with one's original world line, without having to experience the passage of that "missing block" of time on one's own world line.
This means that one's sudden appearance on a new world line would occur with no history on that world line, and one's disappearance from it would then result in no future on it either. Occupants of those other universes containing those alternate world lines would also be able to pop into and out of our universe as well.
In short, sideways time allows movement into and out of every possible world line in every possible universe, and in so doing it would provide philosophers with a new sandbox in which to entertain themselves.
One hopes that some means would exist for keeping cats out of that sandbox...
Think of the puzzle of musical emotion. What does musical notation look like? Among other things, points on a graph. Emotions are like temporal kinesthesia modulo our transit in 2d-time, which is why pure instrumental music can evoke feelings without semantic content expressing emotional events. Music "looks like" those feelings.
More classically, when theologians differentiate between sempiternity and eternity, they are subconsciously tracking the nature of dimensional perception whereby the eternal can see all the points on a given timeline at once, like how 3d-spatial perception allows all the 2d sides of a polygon to be seen at once.
In a Kantian vein, higher-dimensional time is "where" free will operates, free will being the principle of synthesis there just as causality is the principle of synthesis for linear succession. In an Arendtian vein, it is this dimensionality of the will that allows us to break moral timelines using the faculty of forgiveness, and to forge them by the faculty of promising.