Is zakat obligatory on two investment accounts in an Islamic bank, knowing that one of them has completed a full year (hawl) and the other has not completed a full year due to its instability? And how is this zakat to be paid?

If the two accounts originated from a single source of wealth, then they are considered as one wealth, and their full zakat is obligatory. However, if they did not originate from a single source and you acquired them at different times, you have the choice between combining the later acquired wealth with the earlier one and establishing a single hawl (zakat year) for both, which would be the hawl of the first (this is preferable), or assigning a separate hawl for each wealth.

In the second scenario: 1. If each account reaches the nisab (minimum threshold), then its zakat is obligatory along with its profit when a hawl passes over it. 2. If one reaches the nisab but the other does not: If the first reaches the nisab but the second does not, its zakat is obligatory. For the second, there are two situations: it can be combined with the first for the hawl and its zakat paid along with it, or it can have an independent hawl and its zakat paid at its end (and it is not required for it to reach the nisab on its own). If the second reaches the nisab but the first does not, zakat is obligatory for both at the end of the second wealth's hawl, because the first is combined with the second. If neither reaches the nisab individually, but they do reach the nisab when combined, their hawl begins from the date the second wealth was acquired.