You can estimate your future costs with a heat pump using your current utility bill. I plan to switch to a heat pump (23 year old AC and 80% efficient gas furnace... hopefully the HEEHRA rebates become available soon) and have run some of the calculations.
Before jumping into the numbers, a warning that when people talk about "efficiency" of heat pumps, sometimes they mean energy efficiency (heat provided per energy input), and sometimes they mean cost efficiency (heat provided per dollar input). You can see that in this thread: when Nereo says "the heat pump functioned down to -14°F before a resistance coil kicked in" that's energy efficiency. When Sibley says "it's more efficient for a gas furnace when its really cold" that's cost efficiency.
Heat pumps are always more energy efficient than gas furnaces. A gas furnace burns gas and keeps most but not all of the generated heat in your house. Newer models may be 92-96% efficient. My old one is 80% efficient. Heat pumps move heat rather than generate it. At worst, they're 100% efficient (basically just resistance heat). At warmer temperatures, they're much more efficient. For example, in the link below, the MXZ-8C48NAHZ model is 100% efficient at 5F, 270% efficient at 17F, and 375% efficient at 47F (efficiency = COP x 100).
https://www.greenbuildingadvisor.com/question/need-to-know-mitsubishi-heat-pump-cops-at-temps-below-17-degrees-fHowever, gas is usually much cheaper than electricity per unit of energy. It's common at low temperatures for gas to be more cost efficient than heat pumps, while heat pumps are still more energy efficient.
Energy star uses a metric called HSPF (heating seasonal performance factor) to measure heat pump efficiency over a heating season. It's the ratio of BTUs of heat provided per watt hours of electricity used over the heating season. New heat pumps must have a HSPF of at least 7.2 to receive the energy star designation.
https://www.energystar.gov/products/heating_cooling/heat_pumps_air_source/key_product_criteriaIn my case, last heating season I used 430 CCF of natural gas to heat my house, at roughly $1.40 per CCF, or $602 total. I got this from my utility bills by summing the natural gas consumed during the winter months. My furnace is 80% efficient, so 430 x 80% = 344 CCF worth of heat stayed in my house. If I instead had a heat pump with a HSPF of 7.2, then I would have used 4778 kWh to heat my house over that same time period.
(344 CCF) x (100,000 BTU/CCF) / (7.2 HSPF) / (1000 kWh/Wh) = 4778 kWh
1 CCF = 100 cubic feet = 100,000 BTU
My electricity costs last heating season were around $0.12 per kWh, so that 4778 kWh of heating would cost $573. That's a bit cheaper than the cost of natural gas. However, this year natural gas is more like $1.2 per CCF ($516 to heat for a season) and electricity is $0.13 per kWh ($621 to heat for a season), so the heat pump looks less attractive. I also pay $15 per month in connection fees for natural gas. Removing those fees makes the heat pump costs much more favorable.
For that reason, I plan to get electric resistance supplemental heat rather gas back up. The few days a year that I need supplemental heat will be much more expensive with electric resistance heat than with gas, but I won't be paying the other 360 days a year for a gas connection that I barely use.
You can substitute your own usage and costs to estimate how your bills would change with a heat pump. I looked at the entire heating season, but you could look at a single month's usage and get a reasonable estimate for your future monthly bills.
Please let me know if any of my numbers seem off! If I made a mistake, I'd rather find out before I get a heat pump.