On the topic of imaginary-time I apologize for my crude line of questioning, as I'm not well-versed in physics at all but it fascinates me. I was researching the concept of "imaginary-time" and the shuttlecock model of the universe proposed by Hawking and Hartle, and the descriptions I've read explain that the universe started WITHOUT a dimension of time on a vertical axis, but imaginary-time on a horizontal axis which is analagous to a fourth spatial dimension, but that time emerges "later on". I suppose this question is more about the etymology of terms, because I'm confused by these descriptions.
In what sense can there be a "later on" without time if it hasn't emerged as a property yet? Do they mean "later on" in some kind of hyper-time? Like the whole time passes at one second per one second of hyper-time argument? Could they be talking about a block universe where there is no "passage of time" but instead relationships between static bodies along their worldlines that create the illusion of a dynamic universe and time passing? Do they mean that entropy and the arrow of time emerged later on? I clearly have some misconception about what they mean with reference to time, I believe.
 A: To get an intuition for the shuttlecock model, imagine a simplified universe with one spatial and one temporal dimensions. Consider this universe is closed, so essentially it is an expanding circle.
Imagine the Southern hemisphere of a globe with the South Pole at the bottom. In this illustration, the South Pole represents the Big Bang. At any point passed the Big Bang, space in this universe is the latitude circle while time points in the meridian direction. In other words, the worldline of each point on the latitude circle of space is a meridian pointing to the North.
Notice that the specific direction of local time to the future exists everywhere, except for the South Pole. At the South Pole itself, all directions points to the North, meaning to the future. There is no direction to the South at the South Pole = there is no direction to the past at the Big Bang. Therefore a reference to "before the Big Bang" has no meaning exactly the same way as a reference "to the South of the South Pole".
The "imaginary time" concept implies that any direction at the South Pole is horizontal (along the tangential plane). However, in this illustration, the horizontal direction is a direction in space. Therefore, at the South Pole, all directions are spacelike, which is equivalent to "imaginary time". This is true only at the South Pole, which represents the Big Bang with a zero size or duration. However, even one Planck unit away from the Big Bang, the direction in time already exists.
Note that, in the shuttlecock model, there is no incompleteness of geodesic  and therefore no singularity. All timelike and lightlike geodesics cross the South Pole and continue like meridians. The shuttlecock model, as any other, does not describe the Big Bang, but does elegantly explain why there is no such time in the spacetime manifold of the universe as "before" the Big Bang.

Image credit: Quantamagazine
