Here's all the bits and pieces you'll need to build the Fire Lamp.
I'll include links to shops in the UK, and any relevant details, so if you need to source them in a different country, you'll know exactly what you're looking for.
Bear in mind some links may not be working when you read this.
The Ceiling Lamp

This is the light we'll be modifying into a desk lamp - as you can see, originally it's a ceiling fan, hanging by a wire.
The top of the light fitting will become the base, and the ball/arms will become the feature, standing on top of the base.
It can be found on Amazon for about £35 (at the time of writing this article).
It's full name (if you're Googling to find it somewhere else) is the…
ELINKUME Modern Living Room LED Pendant Lights,Creative Adjustable Kitchen Light Fixture,26W Globe Chrome Ceiling Chandelier for Dining Room,Bedroom,Hall Lighting
To build the Fire Lamp, we will simply take this ceiling lamp, remove the wire which the lights hang from, and turn it upside down.
We'll then buy a short rod (see below) to then sit the light fitting so it stands up on the base.
The Stand

Since the original light fitting hangs by a wire from the base, we'll need something solid to stand the ball/arms on top of, so they are away from the base.
You could ignore this step, and have the ball simply sitting on top of the base (it will still be secured by a nut), but we feel it looks a bit nicer being slightly more away from the base.
We'll be going with a 100mm length rod that's threaded inside.
This is available at Lamp Spares for around £5.
The rod has a 10mm threaded hole - it's hollow all the way through.
You might be forgiven for thinking this would be a standard M10 thread, with a standard thread pitch of 1.5mm, however, it looks like all light fittings use a 1mm thread pitch instead.
Knowing this will be important for a couple of other rods, screws and bolts.
Additional Threaded Rod

Next comes another rod, this time with the 1mm thread pitch on the outside.
A total of 30mm in length is plenty.
This is used to attach the stand rod to the base, by getting a threaded bolt/rod on the inside of the base, so we can attach a nut.