I have taken my oldest (4th grader) through the entire elementary Singapore series and now he is into the Singapore New Elementary Math series (NEM1 and 2 cover beginning algebra and geometry). I also have a math degree and am a little obsessed about math curriculum(ok - quite a bit. Have a large pile here of various options).
I think even the base workbook is going to have deeper problems than a curriculum like Alex will have. Singapore problems tend to require application of knowledge to unique problems sooner than other curriculum will. I know another family with a GT boy that is testing 4 grade levels ahead, that came back 3 grade levels to start in Singapore. You can race through a few levels this way anyway. I absolutely do recommend the CWP too, but I think even the base Singapore curriculum is going to considerably deeper than most elementary curriculum. We did have an Alex account for a couple months, but I considered more a way to have extra practice. My son did 3 grade levels of Singapore as a 2nd grader (when we started homeschooling, so I started at grade level). And he did 2 levels as a 3rd grader. Which brought him ready to tackle algebra. One downside of Singapore is that 6A/B does cover some pre-algebra but not enough to jump into most rigorous Alg. 1 programs. Which had me looking at pre-algebra programs which all looked really easy and hokey compared to Singapore.
Whether or not you need the teacher's book really depends on you and your child. If you're math comfortable and/or your child tends to work independently, probably not. If you like some hints to sit down and discuss, you might want it. I've heard people say they wouldn't use Singapore without the teacher's guide. I've never owned the teacher books through the elementary series.
We've also been very happy with the NEM series (1st book anyway). It is extremely challenging. It's not quite the workload of the AoPS books (art of problem solving), but they have plenty of challenge and depth. We're moving fairly slowly! But had we taken AoPS, we'd be crawling and my son would probably be frustrated with the speed and not getting to new concepts faster. He has no problem with concepts, but is not computationally the fast kid around.
