My 406StDT did the same thing after the battery went flat overnight.
Consensus was the HP Pump and was told it would be $5000 to repair, however, I took it to a really good Peugeot tech and he diagnosed it as faulty Coolant temp sensor.
He reset the codes and it was fixed.
Replaced the sensor and it's gone like a charm ever since.
People tend to jump onto the pumps much too quickly in my opinion, when it's only being told what to to do by the ECU that's getting bum info.
Hope that helps
HDi Hunting from cold start
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