I currently work for a very small company (15 persons total). I am not happy in this company, so I went and found a new job at a big ICT consultancy firm. At my current job, I have a 2 months resignation period. For my new job, I have already agreed to start working after this resignation period, but I haven't signed the contract yet. Now, within the last week, another colleague also indicated that he would also be leaving the company, and another colleague became very ill and will be gone for a long time. So, within two months, this company will have lost 20% of its work force... This is causing the company some problems with promised deliverables. So my boss has now asked me if I could stay a few months longer at this company, even though I already made promises to the other company. Should I accomodate him in this?
If you wish to work longer, you work longer. If you don't wish to work longer, you don't work longer. It's up to you. But if you are leaving causes your old boss problems, that are his problems, not yours. You are leaving. You have zero obligations.
Make sure that whatever you sign doesn't make your notice period start again.
"Burning bridges" is generally overrated. If your boss unexpectedly (for you and for him) finds himself in a position where he wants you to stay longer, it is obvious to any reasonable person that you might not be in a position anymore where you can do this. You might have given notice for Friday, and signed a contract to start work elsewhere on Monday. If your boss holds a grudge because of that, then it was a bridge that needed burning.
Worth repeating again and again and again: The order to do things is: Look for new job. Get a job offer. Check what your notice period is. Sign a new contract to start just after the end of your notice period. Give notice to the old company.
It's ultimately up to you to decide, you of course don't have to do it but you have to face the very real possibility that you may be burning bridges if you don't. Whether or not this will be a problem you need to figure out. 2 months is a very long notice period, I personally think that's plenty of time for them to get a replacement, and asking for a couple months more is past ridiculous.