Sounds like you were lucky as you only got a warning which later turned off. As far as I know, when the temp rises there are two warning stages where the first is the temperature is higher than normal and a warning is issued. The next level is the temperature is beyond safe operating limits and the engine should be switched off.
As you only reached the first warning I doubt that any damage has been done. First check the coolant level. If that is OK then the problem can be related to a faulty sensor, faulty thermostat, faulty water pump or a broken/faulty cooling fan or switch.
Go to the garage and get it checked. It may cost a bit but it's a lot cheaper than waiting for the engine to blow!
Went to Garage the guy just opened the hood and said there is no leakage that is visible - he would need to check in detail might need 4-5 hours to find out what is wrong - that translates to about 800 chf.
The guy also told me to let the warning appear once more to get it checked , if I am not planning to go on a long trip.
This was a garage in Meilen ,ZH.
I have also heard about extending warranty - can I get that for 300 or so bucks and if something happens then it will be covered by that.
Get a guarage to pressure chech the system, it's 5 minutes work, then you will know if there is a leak or not!
I think the warranty company will deny any claim, they will work out you bought the cover after you had a problem. Not sure how insurance fraud is delt with here, could be nasty!
There is a hidden menu for the E60 to access the KTMP(koolant temp). Go to any BMW message board and you will find the answer. You can actually monitor the koolant temp live. For your case, it could be that your thermostat was stuck closed, therefore the koolant temp was hight. If the car has close to 80k km, I won't be surprised that the thermostat is on its way out.
Go to a BMW garage, connect the BMW diagnostic computer, and read the vehicle fault history. It will tell you a lot more in 10 minutes than looking around under the bonnet will.
These cars are super intelligent, with lots of diagnostic info.
BUT you need to have the original BMW diagnostic computer to read all of the info - so go to a proper authorized BMW dealer, or an independent BMW garage with the right equipment. Leave those small town garages alone.