I think there’s two concepts being conflated here – one is the familiar GR gravitational time dilation, i.e. the ‘slowing down’ of time due to gravitation (and hence, the local mass/energy density). This is a real enough effect, however, as Exapno already pointed out, time always passes at a rate of one second per second to the local observer, so you don’t observe any difference when you are actually in a place where time runs slower; any difference is only obvious to an external observer, of which, if the time is slowed throughout the whole universe due to the higher energy density closer to the Big Bang, there aren’t any.
The other concept is that of imaginary time, which I believe Hawking refers to a couple of times in Brief History (it’s been a while since I read it). I’m not really sure how best to explain this, though it’s not, in principle, very difficult. I’ll have to take a bit of a round-about approach to this.
In our familiar, three-dimensional space, the (square of) the distance of two things is given by the equally familiar Pythagorean ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup]. (Strictly speaking, the 'd’s mean that this is an ‘infinitesimal’ distance, from which an actual macroscopic distance may be obtained via integration, but this isn’t really important to the topic at hand.) This is what characterizes our 3D-space as Euclidean.
Now, some time after Einstein formulated his special theory of relativity, Minkowski realized that it can be very elegantly and easily expressed in a space where the (squared) ‘distance’ between two events works out to ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup] - dt[sup]2[/sup], incorporating the time coordinate t and thus giving rise to the concept of ‘spacetime’.
This expression can be translated into the familiar Euclidean one by performing what’s called a Wick rotation – essentially, you substitute τ = it (i being the so-called imaginary unit satisfying i[sup]2[/sup] = -1 – this also explains calling the operation a rotation, since, in the complex plane, the imaginary numbers are on an axis perpendicular to the real number line) to obtain ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup] + dτ[sup]2[/sup]; this is what Hawking means with ‘imaginary time’. The reason for doing this, basically, is that it sometimes makes calculation simpler, and you can analytically continue the results from your ‘imaginary time’ calculation back into ‘real’ time.
In a sense, such an imaginary time spacetime now is nothing else but a (four dimensional) ordinary (as in, Euclidean) shape, and we get a degree of intuitiveness from that – we can imagine spacetime as being, for instance, a sphere, with the imaginary time coordinate corresponding to degrees of latitude, which is basically where that whole ‘before the Big Bang = north of the North Pole’ notion comes from; this also shows in an analogical way how to get rid of the singularity at the Big Bang – after all, the North Pole on Earth is a perfectly regular point, as well.
Somewhat more specific, the Wick rotation is used especially heavily in quantum theory, turning a Minkowskian path integral into an Euclidean one, which has the advantage that certain divergences (singularities) can be avoided (path integrals, I should perhaps note, are essentially an alternative formulation of quantum mechanics, replacing the notion of a unique trajectory for some particle/general system with a summation over infinitely many possible trajectories). Hartle and Hawking brought forward a proposal to compute the wave function of the universe using this formalism, called the ‘no-boundary proposal’, which describes a universe that, like a sphere, has no boundaries and thus, no singular beginning (or end).
I hope I got this at least approximately right; I’d welcome anybody with more expertise chiming in.