How close would another aircraft have to fly to warrant a sudden big altitude drop to avoid collision

Most modern aircraft are equipped with TCAS (Traffic Collision Avoidance System) which is a last resort to avoid collision if all other safety nets have failed and two aircraft have gotten too close.

A TCAS can generate an RA (Resolution Advisory) which is a distinct message instructing the pilots what to do - typically either "CLIMB" or "DESCEND". When an RA is generated, the pilots have to comply with it immediately, regardless of all other instructions, unless it directly puts the aircraft in danger. In case of two conflicting aircraft, the TCAS system of each aircraft will communicate with the other, so for example, for two flights on converging courses, one will get a "CLIMB" instruction and the other a "DESCEND".

As for timing:

The maximum generation time for an [...] RA is 35 seconds. The time scales are shorter at lower altitudes (where aircraft typically fly slower). Unexpected or rapid aircraft manoeuvre may cause an RA to be generated with much less lead time. [...] The effectiveness of an RA is evaluated by the ACAS equipment every second and, if necessary, the RA may be strengthened, weakened, reversed, or terminated.

From SKYbrary

Essentially, this means that an RA will only be generated if two aircraft are calculated as being very close to colliding (within 30 or so seconds of collision). Note that the actual distance between the two aircraft does not matter in this regard - calculations are based on "time until impact". Two aircraft on slightly converging courses, closing slowly, can get much closer before an RA is triggered compared to two fast aircraft meeting head on.

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As for the specific incident you are talking about, it is unclear if a TCAS RA was generated. Normally, fighters intercepting an aircraft will turn off their transponder (at least partly) to disable TCAS, with the specific purpose of avoiding such alerts (since they intentionally want to fly close to the other aircraft).

Even if we consider the "1000 metres" claim to be correct, it is not possible to say if that is "too close" or not. If we are talking 1000 metres vertically, that is more than 3 times the normal required separation of 1000 feet. However, 1000 metres horizontally would, in many (but not all) cases, be too close.

So, for instrument flight rules (IFR) flghts, so every airline flight, the minimum vertical distance is 1000ft, so about 300meters. Ofcourse we can't know if the fighter was on an IFR plan or not, or what it was doing up there

That said, are there any alarms or warnings that start going off when another aircraft gets so close to warn the pilot of possible collision

Yes, if both the planes have their transponders on there's a system, called TCAS that, if the planes are in a collision route will thell the pilots what to do to avoid crashing into each other, the pilots MUST do what the TCAS tells them to. But most of the time for IFR flights ATC is responsible for separation between planes