Used to do a lot of weightlifting. The argument is that by using free weights you are using natural movements, constantly having to work on correcting form and posture. Increasing body awareness, core strength, hitting muscles from all different angles etc. But a couple of lumps of metal on a stick can be a bit uninspiring.
Multigyms do look really cool and provide a very efficient way of doing a whole routine in small space. Didn't look at the prices but am guessing from peoples reactions I would need them to come with an on suite bathroom. Think is all about what motivates you to keep going every day. I always think about that Rocky film, lol. Ivan Drago with his high tech gear.
A really cheap alternative is to look at the growing literature on body weight exercising. You can get get some awesome workouts that end up with you pulling Jedi like handstands and weird body knotting yoga type poses. If that kind of thing floats your boat.
Whatever you decide hope you have a smooth move and have loads of fun with your gym.