Yes, if you looked at something going into a black hole, it would appear to you that it is slowing down exponentially the closer it gets to the event horizon, until a point where it appears to completely stop (before reaching it).
But in the aforementioned scenario, you are not an external observer but a part of the system going into the singularity (but analogies have their limits).
slowing down or speeding up? I could be mixing things up but I understand that if I was falling into a BH (lets imagine I'm infinity tough and not torn to shreds due to tidal forces) the universe would appear to be slowing-down/getting-smaller until it had dimensions of zero (or to put it another way, time would "stop") at C. An outside observer would see me moving faster and faster and "stretching" longer and longer.
First you have to understand that a black hole in fact, doesn't have anything really special (until you enter the event horizon), except that their escape velocity is greater than the speed of light.
But gravity applies to it like for any other thing.
So let's say you are the object going into the blackhole, and I'm looking at you.
From your point of view, the closer you get to it, the faster you will travel, just like when you get closer to earth, because it's gravity is stronger and stronger.
So you will end up colliding with it (entering the event horizon) in a finite time (where time, is the time YOU feel, your proper time).
But a black hole is so massive, that space and time get stretched to extreme extents. So from my point of view, you'll be speeding up into the blackhole until a certain point, where you'll appear to be slowing down and stretching (the point where your increase in speed get smaller than the stretching of space-time), and the dimmer you'll get (because of redshifting). So i would see you going slower and slower and appear dimmer and dimmer until almost not moving at all, up to a point where I would simply not see you anymore (before you reached the event horizon).
All this also apply to other object like earth, It would appear to an astronaut that time on earth is flowing more slowly than it's own time (in reality it is of course unnoticeable , but it is true).
Also, to clear things up, you are mixing two point of view here, so here is the separation (you are the object going into the black hole):
"the universe would appear to be slowing-down/getting-smaller until it had dimensions of zero"
From your point of view it would appear to be getting smaller, although I don't think it would go as far as dimension 0.
"(or to put it another way, time would "stop")"
It would appear to the external observer, that YOUR time had stopped. But from your point of view it did not stop, nor did their time from their point of view.
"An outside observer would see me moving faster and faster and "stretching" longer and longer."
You would feel you are falling faster and faster.
The external observer would see you stretching longer and longer.
(Although you would also feel like you are stretching, because of the difference of gravity between your head and your feet, but much less than for the external observer)
Although I understand that at C "time stops". But since time and space are the same thing, this is the same thing as "all distances are zero". From the point of view of a photon the Universe has dimensions of zero. This is why FTL is impossible; it's not possible to travel a distance less than zero; it's completely nonsensical. If we consider C a "speed" in the classical "speed of sound" the idea of FTL makes sense. But when considering C as a fundamental limit of space-time FTL makes no sense at all.
"lets imagine I'm infinity tough and not torn to shreds due to tidal forces"
In fact you don't need to be infinitely tough. If you choose a black hole large enough (like the massive black hole at the center of our galaxy), you would just need a space suit and you could go into the event horizon without being pulled apart by tidal forces. You would be absolutly fine (their may be other dangers though...).
