Alstromeria "Royal Star"

There are some that are easily several feet tall Meanie.
If you like alstros google the Viv Marsh site. He's a specialist nurseryman .... if you dare ;-)
He also has a sale on !!!!!

Will do.
How do I go about splitting it (if possible). Some of the roots are as thick as my thumb!

I think they're tough ol' plants Meanie !
Divide by splicing with a spade and then have the 2 spades back to back and prise apart if the rootball's that large.
It must be quite an established plant !

I only bought it yesterday (half price) Louise. The plastic pot was "round" where it was so root bound and I had to cut it open! So, kitchen knife straight through the middle in this case will do the trick?

Why do you want to split it, Meanie ?
These plants always have very think, fleshy tuberous roots, because they're thick doesn't mean it's a bad thing and need separating.
Perhaps you could just prise the roots apart with your hands if you definately want to split it.

Just exploring my options. I have no home for it until spring anyway. As it is, I've just put into a larger pot to grow into. I can always split it when the new bed is ready for planting next spring! It'll be spoilt with a coldframe over the winter now!