Actually, I think he's analogizing from an area of calculus called complex variable theory. You get the same sorts of problems in complex variable theory - singularities in only one component of the function. 'Imaginary' refers to numbers which are multiples of i, the square root of minus 1. So, if you have the very simple function 1/x + i, as x goes to zero, the real part goes to infinity (a singularity) while the imaginary part remains as i.
Quantitles like time and position are usually considered to be purely real. However, a lot of other physical quantities are complex - that is, they have real and imaginary components. Most quantum mechanical wavefunctions are complex; the current in an electronic circuit is complex. I should mention the 'imaginary' part is no less important than the real part; it just so happens these quantities have the same mathematical properties as a number with a real and an imaginary component.
It's certialy worth exploring what the consequences would be of a complex time.
That's really an interesting analogy, RWP. But I don't see the practical application to our present inquiry. (Maybe you could help me with that.)
I had thought that imaginary numbers are "abstractions" mainly used for practical reasons, as a (very ingenious) way to allow square roots of negative numbers to figure in various types of mathematical equations, which facilitates a wide range of scientific applications; e.g., plotting positions in modeled spaces, etc.
In other words, It seems imaginary numbers facilitate what we already know we want to get done -- they are means to an end. But what is the end, or goal, or purpose? The need to posit imaginary numbers seems to suggest that we already know what that end or goal or purpose is before we begin. So, by analogy, is there a "cosmic knower" who would use imaginary numbers in this manner, and thus produce a singularity? (Is Hawking the "cosmic knower" in this sense?)
Do imaginary numbers really figure in nature -- that is, are they discoveries by man of what is already "there"? Or are they inventions or artifacts of man, useful tools or "machines?"
More questions than answers, as usual, RWP! Thanks so much for writing.