Is it normal for a fast prime to exhibit purple fringing in middle of the image?

I have just bought a Nikon 50mm f/1.8D prime lens.

I took this picture (leg of my glasses frame on a mouse, and a LED desk lamp is on top of it) : f1.8, 1/80s, ISO 200

The purple fringing is right in the middle of the picture. Does it happen very often to this lens?

Answer

This is called axial chromatic aberration (or longitudal chromatic aberration) and it's fairly common with large aperture lenses. It appears as a purple halo around objects that are closer than the plane of focus and as a greenish one around objects that are further, regardless of where they're located in the frame. It often shows up around highlights, like in your example.

This is normal for your lens and will go away when using narrower apertures.

You might be more familiar with lateral chromatic aberration which gets stronger near the edges of the frame but it's absent in the centre. Lateral chromatic aberration is most noticeable around in-focus edges far from the centre of the frame and doesn't go away completely even with narrow apertures.

Lateral chromatic aberration can be corrected very effectively by post-processing the images ("Remove chromatic aberration" checkbox in Lightroom). Unfortunately axial chromatic aberration cannot.

Lightroom does have a "Defringe" setting that can be useful to reduce the effect. But all it does is remove certain colours (settable by the sliders) around edges. It's not smart about it and it cannot distinguish between actual colour detail and chromatic aberration. It can cause artefacts, so use with care.