My employment contract is ending in 4 months, and I'm unsure if the company will offer a new contract thereafter. Should I start looking out, and be open to my boss about my job search?

I would start looking out for new jobs because of the uncertainty (esp with the current economic climate), and while I wouldn't outrightly inform my boss about it, it's something I wouldn't be private about as well. Keeping a lookout for jobs and what's available would give you a gauge of what's out there in the market now which I think would be really useful in an event where the contract does not renew for your current position.

I think it's also important to have an ongoing conversation with your current boss with regards to any upcoming plans/decisions they may have so that you are able to make prior arrangements with regards to your career planning.

All the best! :)