10-10-10 is general purpose balanced fertilizer (since the nitrogen-phosphorus-potash content is the same)that can be used on clematis. I have used it myself. I generally prefer something that is slow release fertilizer regardless of what I use.

Bloom booster fertilizer is a fertilizer that generally means it has a higher middle number and refers to the amount of phosphorus in the mix. Phosphorus is good for root development which in turn leads to better flower production.

I have used a wide variety of fertilizer types on my clematis over the years including 10-10-10 slow release fertilizer, osmocote, compost, alfalfa pellets, nothing but compost, an Espoma organic fertilizer, and probably others that don't come to mind right now. I haven't found one type any better than the others and generally just use what is readily available. I do however mulch my clematis with a wide variety of organic materials including straw, grass clippings, shredded wood mulch, and compost (when it is available) and think that depending on the nutrient content of your soil, that the breakdown of the organic mulch does much more good than any other additive that I have used.