As mentioned above, without D1, C1 will just charge up and no current will flow.
There should be a 1M discharge resistor connected across C1, so it isn't a shock hazard, when the mains is disconnected.
A fuse, is required to protect against fire, if C1 fails.
A diode can be saved, if D1 is removed and the zener, D3 is move in its place, but the output voltage will be reduced by a diode drop, about 0.7V.
Consider increasing C2 to 22µF, for better smoothing and to absorb the current surge, rather than it passing through the zener diode.
Warning: everything connected to this circuit should be treated with the same precautions as mains voltage. The entire circuit should be placed in an insulated enclosure, made from a flame retardant material. If the enclosure it metallic, it needs to be connected to earth/ground.