In general for CW multipliers, you *dont* need the output to be pulled below ground, but it does need to be pulled down to ground to discharge to first cap. I.e. a totem pole output will work, but an open drain/collector/emitter/source output wont. For example, you can use a 555 (which has a surprisingly beefy output) to drive a 3 stage multiplier to get a 10-15v output for diving mofet gates.
But yes, the thread starter needs another diode and smoothing capacitor on the output for his circuit to work.