You should absolutely keep your US bank, brokerage, investment, IRA, or retirement account when you move overseas.
Opening financial accounts abroad exposes you to additional risks and a lot of reporting. This is paperwork not only for your tax return, but you may need to complete the FBAR (Foreign Bank Account Report). The FBAR is not exclusive to just bank accounts, but also applies to life insurance policies with a savings component, mutual funds, brokerage accounts, etc.
Assuming you're not a retiree, your tax return will be much more subject to scrutiny if you have a lot of assets abroad. With FATCA (Foreign Account Tax Compliance Act), the US government will have access to foreign bank account balance and income information of US citizens living abroad.
Therefore, working expats are now much more on the radar screen than they were before. There are also reporting requirements. Given the fact that the Treasury Department and the IRS now have more access and could get greater information on a foreign bank account than they do a US bank account, in some respects, expats need to file their tax returns even more accurately.
For example, I have a client who stayed in Mexico who forgot about US $15 worth of interest income on a bank. He got a letter from the US government saying that he forgot to report that $15, which is amazing. Gone are the days when people actually had to look at papers. Now, the computers are doing all the reconciliation.
Foreign banks and foreign financial institutions are sending account information to the US Treasury Department. They are now able to reconcile that with the tax return and the FBAR. Expats filing their income tax returns must be much more accurate.
There are all kinds of risks of investing abroad such as country risks, political risks, and "know-how" risks. You may be subjected to scams. You can easily invest in foreign markets through US accounts. I would do this.
Leave accounts in the local country for liquidity purposes. For example, I would need funds in a bank account to keep a small amount in the money market or a CD (certificate of deposit) because I need an account to earn interest while I spend the money down.
For the most part, I would maintain most of my assets in the US.
I have a client who has been living in Guatemala for a long time. She has let go of all connections in the US, including her US bank accounts. The US has the PATRIOT Act. She has severed her ties to the US completely, not just for residence purposes, but also for financial account purposes. Now that she wanted to open a US bank account for another reason, she is having a ton of difficulty even though she is a US citizen because there is no recent record of her.