A bad neutral isn't going to trigger a brownout that reproduces reliably in some random gzip code when you're booting fully into Windows, on a modern CPU with fine grained frequency/voltage scaling, on a battery-powered laptop, powered by a switched-mode power supply with tons of capacitance in every single layer on the way to the CPU.
The most plausible explanation I can think of would be if the return current is flowing through the monitor and EMI from that is disturbing things, but 60Hz might as well be DC to modern signal rates, so that's tantamount to blaming witchcraft.
That is exactly what bad neutrals do (produce brown-outs). They produce brown-outs because the potential between the "hot" side and neutral is < 120 V, because the neutral is > 0 V.
Yes, it'll produce a brownout at the outlet, but it's highly unlikely that this will manifest as a brownout at any of the component voltages, because there are so many layers of filtering. That <120V goes through a switched-mode power supply which converts that down to 20ish V, which goes to a battery charger IC which charges the battery, which outputs a variable voltage that goes through multiple more layers of monitored DC/DC conversion to generate the power rails. Every one of these steps has circuitry that will panic and send a signal to the main PMIC if it sees a voltage that is out of spec.
My guess would be that the issue was not brownouts, but noise (which could even have peaks well above 230V). The air conditioner has a motor, and motors can be electrically noisy; it should have filtering, but with a broken neutral, that filtering might not have been as effective. Grounding the outlet with the computer (the solution at the end) would have allowed the filters at the computer and monitor power supply to filter out the noise (AFAIK these filters work mostly by shorting high-frequency components of the waveform to the other wire and/or to ground, so they need a good neutral and/or ground to work).
But yeah, these sorts of "bad wiring" situations can have baffling effects. I would love to see someone in one these situations actually manage to get a logic analyzer trace or similar to show what's actually going on with the power supply, together with an investigation (and fix) of its real cause.
Most residential power is dirty to some extent, and AC/DC converters are supposed to be able to cope with a wide range of input distortions. For the problem to have worked its way through all of the safety components in both the laptops power brick and its internal electronics means that the distortions must have been particularly high amplitude, and at frequencies that couldn't be easily filtered out (very high, very low, or both).
Electricians that aren't particularly well trained, or particularly experienced are probably not going to be able to diagnose this kind of problem.
This is really a problem for an electrical engineer.
Unfortunately, in most jurisdictions, engineers aren't allowed to perform these kinds of repairs because they don't necessarily have the certifications to perform work on live wires.
Most electricians can only trace wires and identify visible faults, and replace things that aren't up to building codes.
You can buy a $10 tester[0] that will detect open grounds, open neutrals, and any other wiring fault you'd care to think of. It doesn't take an experienced electrician to use; you plug it in, and see which lights turn on.
What I think is more likely in this case is the A/C turning off. All electric motors induce a current whose direction is opposed to the originally applied current (Lenz's Law). This is called back-EMF or counter-EMF. It's also proportional to the rate of change of the magnetic flux. You can see this when you turn on a motor like a vacuum cleaner, and observe the lights briefly dim. The same thing happens when a motor turns off. Additionally, things like vacuum cleaners or fans have relatively little resistance to them, so when they turn off they can coast down - small rate of change, small impact to voltage levels. An A/C or refrigerator compressor presents a relatively large resistance to its driving motor, so when it shuts off, it shuts off HARD. Big rate of change, big induced voltage spike. I don't know about a small motor like you'd see in a household, but large industrial motors can generate hundreds of volts when turned off. This is usually handled with a circuit called a snubber, but it doesn't generally matter in residential applications so it's rare to be included, as a cost savings measure.
The most plausible explanation I can think of would be if the return current is flowing through the monitor and EMI from that is disturbing things, but 60Hz might as well be DC to modern signal rates, so that's tantamount to blaming witchcraft.