Yes, it is possible and permissable if correctly done. It's called a multi-wire branch circuit (MWBC) and uses a shared neutral.

"There will be two separate breakers on my panel,

The two hots must be on opposite poles--wired at the breaker panel like a 120/240 volt circuit. Otherwise, the neutral is subject to overload. Remember that a neutral is a current-carrying conductor and if both hots are on the same pole, the neutral would have to do "double duty". When the two hots are on opposite poles, however, the neutral carries only the imbalance between the split-phase hots.

Furthermore, the 'two breakers' can't be entirely "separate". In the newest code versions, a two-pole breaker is required. That's to ensure that when power is turned of for one of the circuits (or one leg of the MWBC, if you prefer that terminology), the neutral isn't still carrying current from the "other half".

Run two separate circuits.

You do not want a computer sharing a neutral with an AC unit.

The on/off voltage sags and surges from the AC unit will not be good for the computer.

What type of computer are you using that you want to run a new 20A circuit?

As opposed to what brickeye said, 90% of the worlds residential computers (and plasmas, and other electronics) are powered from the same circuit as AC units, vacuum cleaners, etc. without any issue whatsoever. To just share the neutral will give you even less problems (can you get any less than none?).

"As opposed to what brickeye said, 90% of the worlds residential computers (and plasmas, and other electronics) are powered from the same circuit as AC units..."

Nice invented number.
While house residential systems are normally 240 V for the condenser, and a separate circuit for the air handler on 120 V.

Vacuum cleaners are nowhere near the power hog that a window a/c unit is, and many of the window units recommend dedicated circuits since they pull so much on 120 V.

If you want to run your expensive electronics on a circuit that WILL have voltage sags at turn on of the AC compressor, and voltage surgesz when the conpressort shuts off, go for it.

Remember that surge protection devices have a turn on voltage of over 170 V already so the peaks of the 120 V do not trip them on, and that each and every time a MOV breaks over it is damaged and has less and less power handling capability.

I assume you are trying to save money on wire? If that is the motivation, I'd recommend paying a little extra and just running 2 lines. A dedicated circuit for expensive electronics is a good extra safety feature. You can prevent some potentially expensive problems with a relatively small investment.

Seriously, Brickeyee, I'd like this explained for my own edification.

"If you want to run your expensive electronics on a circuit that WILL have voltage sags at turn on of the AC compressor, and voltage surgesz when the conpressort shuts off, go for it.

How, exactly, does the the 'turn on' of an AC compressor cause a voltage sag on the other leg of a MWBC that would exceed any voltage sag expected from two equally size 'independent circuits' from the same panel? Within the practical parameters of the OP's problem, is this really true?

Plain old volate drop.

When the compressor starts up the there is a voltage drop from the starting surge on both the hot and neutral of the circuit.

The drop on the shared neutral appears as a sag to a load on the opposite leg of the shared neutral circuit since its 0 V reference (the neutral) has a large surge current creating a voltage drop.

The voltage surge occurs when the compressor shuts down and the collapsing magnetic field of the motor drives the line and neutral apart from each other. The voltage has no reference except the neutral, and its inductance will cause the neutral voltage to be driven below zero volts with a nasty waveform that depends on how fast the magnetic field collapses.

It is the shared neutral with its impedance that couples the two sides of a MWBC and allows transients on one leg to appear on the voltage in the other.

terribletom - Just trying to understand the angle you are coming from. Is your argument that the neutrals are going to be connected in the panel anyway? Specifically, the neutral for the computer and the rest of the room is going to either be shared inside the room or at the panel and it shouldn't matter which? Any effects caused by the compressor are going to be felt in the neutral for the PC either way?

Thanks for the reply, Brickeyee, and I'm sorry I couldn't get back to this sooner.

I've read a bit about the problems with harmonics that seem to happen mostly in commercial settings where a bunch of computers or fluorescent lighting is wired on MWBCs on three-phase Wye systems, and the arguments for so-called super-neutrals.

But what I suppose I'm not getting in the context of the OP's scenario is the effect on the neutral when the currents are 180 degrees apart. I would have thought that the neutral would still present a decent enough in-phase 0V reference to the computer's power supply when it's fed by an opposite-pole leg. But if it's inductance that really tends to confound things, that's beginning to make sense to me.

terribletom - Just trying to understand the angle you are coming from. Is your argument that the neutrals are going to be connected in the panel anyway?

Thanks for asking, billl. Well, no, I wasn't theorizing to that extent. I was merely seeking to isolate the problem beyond mere voltage drop (which can, of course, be "felt" by independent ccts from the same panel--more so if it's a small panel and the two hots come from the same pole).

It's specifically the current-carrying characteristics of a shared neutral on a MWBC that I was interested in. So I thought that by distinguishing between circuit types, I'd get a more targeted answer.

Let's say this is more about learning than arguing, eh?

Look at it this way - if this was a DC circuit the voltages would be +120/0/-120 with no load. When the A/C kicks on the voltage might be +118/+2/-120 . So the computer would be seeing 122volts while the A/C would be getting 116volts.

Anyway, I'm with whoever is recommending independent circuits for the A/C and computer.

Run two, it's really no more labor, a little more copper, and you'd have to add some creative splicing/wiring between the two circuits in the room anyway. Bring in one alone for A/C, and the other for the stuff. It's easier to do, I think, and MUCH easier to troubleshoot in the future.

And, unless the atom-splitters disagree, you can land it in the panel where most convenient.

KEN