This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 4 solutions found in 0.178 seconds (with a Bezout bound of 16).
Reference: "Solving deficient polynomial systems with homotopies which keep the subschemes at infinity invariant" by T.Y. Li and X. Wang (pages 693-710).
See also: http://homepages.math.uic.edu/~jan/Demo/lumped.html.
i1 : lumped(CC_53)
o1 = {(.767719 + .328203*ii)x x + (.767719 + .328203*ii)x x + (- 1.76772 -
1 3 1 4
------------------------------------------------------------------------
.328203*ii)x + (.548909 + .109395*ii)x + .0479608 + .888687*ii, (.3301
1 3
------------------------------------------------------------------------
+ .890584*ii)x x + (.3301 + .890584*ii)x x + (- 1.3301 - .890584*ii)x
2 3 2 4 2
------------------------------------------------------------------------
+ (.111291 + .671775*ii)x + .829152 + .669877*ii, (- .767719 -
4
------------------------------------------------------------------------
.328203*ii)x x + (- .767719 - .328203*ii)x x + (- .892482 -
1 3 1 4
------------------------------------------------------------------------
.452966*ii)x x + (.767719 + .328203*ii)x + (- .548909 - .109395*ii)x ,
3 4 1 3
------------------------------------------------------------------------
(- .3301 - .890584*ii)x x + (- .3301 - .890584*ii)x x + (- .892482 -
2 3 2 4
------------------------------------------------------------------------
.452966*ii)x x + (.3301 + .890584*ii)x + (- .111291 - .671775*ii)x }
3 4 2 4
o1 : List
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The object lumped is a method function.