A: Mike McArtor, copy editor for DailyMTG:
In order for a card to be included in a Commander deck all of the colors of its entire color identity must fit within the color identity of your commander. Color identity is defined as "…the set of colors the card is and the colors of any mana symbols appearing on the card." Hybrid mana symbols count as both colors, so, to use your example, Bioshift is both green and blue, meaning it can't be included in a mono-green deck.
The same is true of a split card, where both halves have to be within your commander's color identity in order to include it in your deck, even if you only want to use one side or the other. Going back to your examples, Profit & Loss is both white and black, meaning you can't include it in a Commander deck unless your commander includes both white and black in its color identity. It gets even more restrictive when you look at split cards like Ready & Willing, where your commander must have black and green and white in its color identity to include the card.
Color identity extends to the text box (but does not include reminder text, like that for the extort mechanic), so, for example, Scholar of Athreos's color identity is white and black. Oh, and double-faced cards? Yup, both sides count for color identity, whether because of a color indicator (as on Cloistered Youth/Unholy Fiend) or because of mana symbols (as on Daybreak Ranger/Nightfall Predator).