It must be hermetic, to keep the argon inside and the air outside, for the lifetime of the bulb.
It contributes to the high efficiency, by reflecting both the visible and the infrared light and preventing thus the heat loss from the lamp.
Nonetheless, alumina is less fragile than ordinary glass.
It might be possible to replace the quartz window with some kind of glass window, for a lower cost, but not with any kind of plastic, because it must also be hermetic and it must be usable as a substrate for thin film deposition in vacuum and it should have a coefficient of thermal expansion matched to those of the multiple layers of the infrared-reflecting filter.
I didn't mention this in my other comment because I thought the use of alumina could be avoided, but I suggest you look up the price of some optical sapphire (ie transparent alumina) before you get excited about that. Even for a tiny, flat circle you're looking at around ten bucks. A bulb would likely be harder.
https://www.uqgoptics.com/catalogue/optical-windows/sapphire...
The light exits the lamp through a flat window, which is made of quartz in the prototype, but in mass production it is likely that some kind of alumosilicate glass or borosilicate glass would be fine for it.
If in mass production, I see nothing that would prevent it from being cheaper than a comparable LED lamp.
I am not familiar with the technology used to make the incandescent band made of carbon nanotubes and boron nitride ceramic, but everything else is made with standard equipment that is widely available.