I don't know anything about desert iguanas, just my big green iguana, but I do also have a beardie. My recommendation? Get a beardie. They are permanently happy as far as I can tell, grow to be a wonderful size, just enough to sit on your shoulder or in your hand, and are fairly easy to keep. I love my Tempest, but he is much more work than my beardie, and quite a bit more tempermental. The only downside to beardies is that they brumate, and I did miss my little guy while he was down. At the same time, I'll take a beardie brumating over an iguana's mating season!! It all depends on what you're looking for in a pet I suppose, I get far more satisfaction out of dealing with Tempest and having tamed him up, and in keeping him healthy, but if I'm going to go hang out on the deck and read I prefer my beardies company on my shoulder
Tempest comes out too, but up on the hammock stand where he can perch and sneer down at all 