I tried to find the roots of the 2nd level Mandelbulb (degree 8) function by numerically finding roots starting from a whole mess of random seed points and then removing duplicate results. I got 64 unique answers, and they appear to be situated in a configuration that looks kinda like the Mandelbulb (see attached image below).
The coordinates of these 64 roots to double precision (rounded from higher precision) are:
{{-1., 0., 0.},
{-0.8117449009293668, -0.3909157412340149, -0.4338837391175581},
{-0.8117449009293668, -0.3909157412340149, 0.4338837391175581},
{-0.8117449009293668, 0.3909157412340149, -0.4338837391175581},
{-0.8117449009293668, 0.3909157412340149, 0.4338837391175581},
{-0.766044443118978, 0., -0.6427876096865394},
{-0.766044443118978, 0., 0.6427876096865394},
{-0.6234898018587335, -0.7818314824680298, 0.},
{-0.6234898018587335, 0.7818314824680298, 0.},
{-0.4776208980552355, -0.5989176626001069, -0.6427876096865394},
{-0.4776208980552355, -0.5989176626001069, 0.6427876096865394},
{-0.4776208980552355, 0.5989176626001069, -0.6427876096865394},
{-0.4776208980552355, 0.5989176626001069, 0.6427876096865394},
{-0.20048443395120957, -0.8783796973249267, -0.4338837391175581},
{-0.20048443395120957, -0.8783796973249267, 0.4338837391175581},
{-0.20048443395120957, -0.0965482148568969, -0.9749279121818236},
{-0.20048443395120957, -0.0965482148568969, 0.9749279121818236},
{-0.20048443395120957, 0.0965482148568969, -0.9749279121818236},
{-0.20048443395120957, 0.0965482148568969, 0.9749279121818236},
{-0.20048443395120957, 0.8783796973249267, -0.4338837391175581},
{-0.20048443395120957, 0.8783796973249267, 0.4338837391175581},
{-0.17364817766693036, 0., -0.984807753012208},
{-0.17364817766693036, 0., 0.984807753012208},
{-0.10826786788668456, -0.13576361217320798, -0.984807753012208},
{-0.10826786788668456, -0.13576361217320798, 0.984807753012208},
{-0.10826786788668456, 0.13576361217320798, -0.984807753012208},
{-0.10826786788668456, 0.13576361217320798, 0.984807753012208},
{-0.049515566048790434, -0.21694186955877906, -0.9749279121818236},
{-0.049515566048790434, -0.21694186955877906, 0.9749279121818236},
{-0.049515566048790434, 0.21694186955877906, -0.9749279121818236},
{-0.049515566048790434, 0.21694186955877906, 0.9749279121818236},
{0., 0., 0.},
{0.03864035467425736, -0.16929445530699877, -0.984807753012208},
{0.03864035467425736, -0.16929445530699877, 0.984807753012208},
{0.03864035467425736, 0.16929445530699877, -0.984807753012208},
{0.03864035467425736, 0.16929445530699877, 0.984807753012208},
{0.1387395330218428, -0.17397387167523584, -0.9749279121818236},
{0.1387395330218428, -0.17397387167523584, 0.9749279121818236},
{0.1387395330218428, 0.17397387167523584, -0.9749279121818236},
{0.1387395330218428, 0.17397387167523584, 0.9749279121818236},
{0.15645160204589237, -0.07534312061707779, -0.984807753012208},
{0.15645160204589237, -0.07534312061707779, 0.984807753012208},
{0.15645160204589237, 0.07534312061707779, -0.984807753012208},
{0.15645160204589237, 0.07534312061707779, 0.984807753012208},
{0.17046092493487977, -0.746838109568473, -0.6427876096865394},
{0.17046092493487977, -0.746838109568473, 0.6427876096865394},
{0.17046092493487977, 0.746838109568473, -0.6427876096865394},
{0.17046092493487977, 0.746838109568473, 0.6427876096865394},
{0.2225209339563144, -0.9749279121818236, 0.},
{0.2225209339563144, 0., -0.9749279121818236},
{0.2225209339563144, 0., 0.9749279121818236},
{0.2225209339563144, 0.9749279121818236, 0.},
{0.5617449009293668, -0.7044058256496909, -0.4338837391175581},
{0.5617449009293668, -0.7044058256496909, 0.4338837391175581},
{0.5617449009293668, 0.7044058256496909, -0.4338837391175581},
{0.5617449009293668, 0.7044058256496909, 0.4338837391175581},
{0.6901821946798448, -0.3323742273106898, -0.6427876096865394},
{0.6901821946798448, -0.3323742273106898, 0.6427876096865394},
{0.6901821946798448, 0.3323742273106898, -0.6427876096865394},
{0.6901821946798448, 0.3323742273106898, 0.6427876096865394},
{0.9009688679024191, -0.4338837391175581, 0.},
{0.9009688679024191, 0., -0.4338837391175581},
{0.9009688679024191, 0., 0.4338837391175581},
{0.9009688679024191, 0.4338837391175581, 0.}}