Depth of roots will vary depending on soil type. They are often shallow to mid rooted and fairly wide spread. The roots will probably be wider than the foliage. They dont like root disturbance and this is the main cause of death. Winter (now) is best if you do want to move it but not when frosty.
Just a point, there was a monkey puzzle tree on a TV prog the other day, and the presenter said that some of them are listed and therefore cannot be chopped down. I'm pretty sure she was on about the larger older ones, but it might be best to check with your local council before moving it.
They are very wind-firm and rarely get blown down, so be prepaired, the roots may extend further than might be expected.
I successfully transplanted one a couple of years ago and it's doing well, the addition of plenty of well rotted leaf mould forked into the planting hole,seems to have helped it settle in. I also give it an annual mulch of bark chip which keeps the weeds down if nothing else. Good Luck Tbird+