Here's a picture that I've posted elsewhere that is a classic example of an opportunity for moire, taken with my E-M10, which has no AA filter. This picture is straight out of the camera. No moire on the grill....
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I don't think they're necessary any longer, but engineers may think otherwise. It always seemed to me that moire was a result of sensors that didn't have sufficiently dense resolution to capture high contrast, fairly frequently alternating pixel brightness. I think (although this is just a gut feeling) that with much higher res sensors available today with pixels of sufficient physical size, this has pretty much disappeared as a major concern. In the old days of digital photography moire was a frequent complaint and so the manufacturers likely dealt with it by adding "anti-aliasing filters", even though no-one knew how the hell they actually worked (probably by optically softening edges). My first digital camera was a Canon Digital Elf with a whopping 2 megapixel resolution. Yes, I had moire.
I don't know if you see it on your particular monitor, but the notebook that I'm using to type this has a 1920 X 1080 screen. When I scroll the picture above I see moire, but when I stop it disappears. Presumably this is because the resolution and refresh rate are not sufficient to accurately render the detail of the picture as it's moving. I'm guessing that if you have a higher res monitor with a faster refresh rate, you may not. If you're using a CRT monitor ("a what?", ask the youngsters) you almost certainly won't see any moire. What I see is effectively moire. I think this is the same basic cause of moire in captured pictures; insufficient resolution combined with insufficiently large pixels on the sensor to capture and present the image at a sufficiently high resolution that my eyes can't see the rough edges that are actually there and the weird patterns that result when they start to mush together. That's just my theory, but my real-world observations are consistent with it.