I can think offhand of two, 20driver.
1. B747SP ex-JED (2 engines failed on same side) at 800agl, aircraft dumped (ASAP) and returned.
2. B747 ex-BAH enroute ATH
Over Saudi, one engine flamed out, another on final at ATH, and just prior to parking on stand, a third wound down.
Severe fuel contamination...ex-BAH, altho the airport folks there would never admit to same (why are we not surprised?).
Now, I have spent over thirty years in command in 3 and 4 engine heavy jets (same rules for both...ie: continue on, diversion not necessary with an engine failure), but I have to wonder...at what point do you continue to stretch your luck, before Murphys law takes affect.
BA ex-LAX, where the failure occured just after takeoff, for a planned ten hour flight to LHR would seem to me to be stretching your luck to the extreme.
An engine failure over well up over Canada, yes, continue might be prudent.
But, just after takeoff?
Well, not for me, certainly, regardless of aircraft type.
There are those here who think I might be too cautious, after all these aircraft are designed for flight on three.
Yes they certainly are.
BUT, there has to be a point where enough is enough, no matter how reliable the 4 engine aeroplane is.
And, an engine failure/shutdown just after departure seems to me to call for a diversion...or return.
Certainly prudent, in my view.