Quick question I have one Origin ID I can log in on both computers I have. Now if I were to purchase a Sims expansion pack would that carry across because it was bought with the same account? And If I were to get a new pc/mac in the future would I be able to play by logging into my Origin account and starting a new game that way with new all the extras I will have purchased?
Yes - the expansion pack was purchased against the account, not the computer
Because it was purchased against the account, to install it on another computer or any future computer you may get, simply install it from under "My Games" in the Origin client.
Note that with downloadable content, rather than expansion packs, these are all grouped together under the main game. To install these, you need to right click on the game and select "Show Game Details" to view the DLCs you have purchased.