The question of endings is always very important, as how a text ends clearly has a massive impact on the reaction of the audience. It would be tempting to try and make this play a traditional "happy ending" and have Desdemona and Othello walking off together arm in arm towards a beautiful sunset, but part of the power and intensity of this text lies in its tragic ending and the way that Othello is undone by his own jealousy and inability to control his thoughts. This is something that Othello himself acknowledges in his final speech in Act V scene 2, just before killing himself, when he describes himself as follows:

Then you must speak
Of one that loved not wisely but too well,
Of one not easily jealous but, being wrought,
Perplexed in the extreme...

If Othello had not killed Desdemona and then himself at the end of the play, the impact of the play and in particular the dangers of jealousy and being too open to manipulation would not have been highlighted so strongly. Othello needs to bear some sort of punishment for the way that he is able to be manipulated so easily by Iago, and Iago is not able to take all the blame. This is something that arguably could have occurred without the need for Desdemona's death, but on the other hand this is something that highlights Othello's true self and his real failing of how he acts when he is possessed by jealousy. The play seems to suggest that this fatal flaw within Othello would always be there, waiting for somebody else to identify and exploit it. A happier ending where this would have been brushed under the carpet might have been seen as putting off the inevitable.