[ahazred8ta]

That's an urban legend. In real life physics, the traveler is being pulled forward by the black hole's gravity, faster and faster, and as it covers the last few miles it's traveling very close to light speed and crosses the last few miles in a tiny fraction of a second. The time dilation is not enough to cancel the increased speed. There's no wall of molasses. An object falling in goes faster, not slower. Objects falling into the black hole do not actually hover just short of the event horizon, despite appearances.

[rbanffy]

Shouldn’t we at least see a visible Doppler shift at least.

[ahazred8ta]

A typical stellar-mass black hole of 3.3 solar masses has a radius of 10 km, 10,000 meters. At 100 meters outside the event horizon there is a 10:1 time dilation. At 1 meter outside there is a 100:1 time dilation. At 1 cm outside there is a 1000:1 time dilation.

But because you're traveling at almost the speed of light while you're falling in, it only takes you a few microseconds to fall past these points and get to the event horizon. If there was a mirror 1 mm outside the event horizon, a photon could start from 1000 km away, fall in, hit the mirror, and climb back out in 1/100th of a second, even allowing for time dilation. The spaceship will only spend a very tiny fraction of a second in the high time dilation area, even as seen from outside.