Well, I differ there. If someone gives me a compliment, I appreciate it. Usually it is about a hair cut, or a choice piece of clothing. I never took it as a sexual advance. However your post bring up an important subject.

There are differing opinions here. Why is one opinion more justified than the other in the work place? For example: Why should I not be able to give a genuine complement about let say a woman's shoes? The work place is so filled with terror to the point where some companies policies are that a member of the same sex must be present to do simple coaching tasks. Is this not absolutely ridiculous?

There's quite a difference between complimenting someone's shoes vs. saying she looks cute.

Would you feel the same if someone said, "nice haircut" vs. "you look cute with your new haircut"? I highly doubt it. And further, try imagining it coming from differing genders, and differing sexualities.

One of the compliments objectifies, and that is why it doesn't belong in the workplace.

"Attractiveness" implies compliments that are suggestive. But, personally, I veer away from anything along those lines because I do not know what my colleagues are comfortable with. It is not my place to assume.

Out of curiosity, what do you think about appraisals of a person's other qualities? Maybe it's because I'm bad at the whole compliment business anyway, but I'd rather have a female coworker notice that I've started going to the gym (harmless, fun) than to be called smart or a good organizer (ohmygod this actually matters at work, power games!!).

You have to consider the power imbalance due to gender norms. In our culture, men typically pursue women romantically. This is not always the case, but it is the norm. Hence, in a social setting where it is acceptable to pursue someone romantically, a woman complimenting a man's appearance has the implied meaning "I would be receptive to your romantic advances." [1]

However, in the same social setting, when a man compliments a woman's appearance, the implied meaning is "I am pursuing you romantically." And this is fine in social settings, otherwise our species would die off.

But it's not fine in the work place. You feel that a woman at work giving you such a compliment is "harmless fun" because if it was a normal social situation, you would have the power: it's up to you to pursue. But for many women it is not "harmless fun" because it's similar to the social situation, and they feel romantically pursued, which is not okay at work.

[1] Please note I do not consider this a promise, so let's not confuse this with "asking for it." Human communication is subtle, and such implications are how consensual "courtship" happen. Of course, once there is a rejection, whatever implications one thought were there don't matter.