I noticed a purple-ish outline around my photo, however, I couldn't remove it through the Lightroom Defringe option. I have tried the fringe selector and playing around with the slider, nothing seems to work.
Purple fringing, also known as longitudinal chromatic aberration, is not the same as lateral chromatic aberration which can easily be compensated for in software (though not fully corrected since each primary color of the sensor spans a range of wavelengths). Lateral chromatic aberration results from different wavelengths of light being projected with slightly different scale to the sensor, typically leading to red/green fringes at the outer parts of the image.
Longitudinal chromatic aberration means that the focus plane for different wavelengths is different. The effect occurs over the entire image and cannot be compensated systematically. Some image processing programs still offer tools to detect and correct such fringes. The typical scenario for purple fringes is sharp edges (such as branches) before the sky. Since the sky contains a high amount of short wavelength visible (and partly UV visible only to the camera) light, the edges may be out of focus for short wavelengths more than for other wavelengths and have purple from the sky bleed over.
Now this may contribute to the basic purple hue you have in your edge, but in your example image there is a lot more going on. It appears like the arm is considerably out of focus and you applied generous amounts of sharpening and likely adaptive noise cancellation. This leads to considerable halos around the edge that also tend to spread out what may have started as a purple fringe from longitudinal chromatic aberration.
Starting from a photograph that is better in-focus will likely make it easier to figure out what is happening, and it might also make some of the image processing corrections be more successful at detecting and masking the fringing.