The economics of both fossil fuel and heat pump systems depend on a whole lot of variables. Fuel prices fluctuate, as do electricity rates. In many areas, they are tied to each other by the power utilities, especially those burning coal or natural gas to generate line power. This is only one piece of the puzzle.
The overall comfort level provided by the system employed is another important factor to consider. Gas and oil furnaces tend to run more economically if the burn time is minimized. This is accomplished by using relatively short blasts of hot air, followed by a "quenching" of the heat exchanger, then shutoff until the next call for heat. Many mini-split models use a different scheme, providing "merely warm" air for a longer period of time during each call for heat. The more elaborate models (variable-speed inverter type) actively manage the discharge air temperature (to a lower outlet temperature) to maximize unit efficiency. Generally, the more gradual the temperature changes, the more efficient a heat pump unit is.
This operation is not appreciated by many owners who have become accustomed to the "hot air blast" method employed by the old-fashioned fossil fuel heaters. This is why some owners are never completely satisfied, with regard to comfort levels, of heat pump systems. Since the heat pump doesn't provide noticeably hot air from its discharge vent, it can't be heating well...
Some manufacturers have integrated "hyper heat" or "turbo" modes into their systems to attempt to replicate this initial blast of "hyper-heated" air. Regardless, it's not the same effect as a fossil fuel unit. Some users are satisfied by this mode of operation, at least until they figure out it affects the heating bill (in a bad way) to run the units in this mode. As xringer described with his sanyo units, this "turbo heat" mode stresses the heat pump and can lead to longevity issues.
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