Having done quad in the past a little "there I was" story". I just once had to shut an engine down in the cruise (due to low oil pressure - the engine was running fine). We were over a part of the world where whilst we had alternates with nice long runways they would have been a nightmare from both a political and a logistical /passenger handling POV. Head office and the engine manufacturers were on the case before we'd finished drifting down, thanks to the wonders of datalink and satcom, all our onward alternates were pretty much CAVOk so after due consideration we plodded on a scant three engines for several hours to destination. Safety report filed, never made the papers, didn't even make the company's horror comic
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I suspect if we'd dumped fuel and landed on the nearest suitable piece of concrete, which seems to have been the knee jerk solution some are suggesting it would have taken days to get the passengers out and weeks to move the aircraft.....but I guess we would have had plenty of time to debate the merits of continuing. So FWIW so would I do the same again in the same circumstances if I was on a quad - yes, absolutely.
Now I fly a twin as Hi_Tech has said the thinking is somewhat different, but we're discussing engine failure on a quad, right?.........