The Anglepoise Apex PL was manufactured between 1985 and 1997. At the time it was developed around a new bulb type called the PL bulb in an 11 watt configuration developed by Philips. It used a dedicated G23 lamp holder and required a choke / transformer in the plug.
One of the most common failures we have noticed recently is in the choke plug where it just fails, which isn't really surprising after 20-30 years of usage. We used to have replacement plugs but these are no longer stocked, and we can no longer get hold of any.
We have however found a solution to this. It uses a new bulb that utilizes modern LED technology that mirrors the dimensions of the original 11 watt PL bulb. But it is mains voltage and doesn't therefore require a choke / transformer in the plug. In fact, it must be replaced by a standard plug with a 3 amp fuse for this to work.
The other advantage is that the light source gives off a vastly improved quality of light and brightness., and you can with some suppliers choose the colour temperature of the bulb. (2700k, 4000K or 6000K)
The bulb I found that worked was purchased from an external website here (product may change from link) and you will need to find a plug with a 3 amp fuse.