I think the problem is that the default value of ratepsilon is too small; try this:

(also do this same with ratepsilon : 1.0e-8)

(%i1) load(hypergeometric)$

(%i2) ratepsilon : 1.0e-18$

(%i3) m : matrix([2621.4397,-7823.3599,-1111.2726],[-7823.3599,23347.842,3316.4543],[-1111.2726,3316.4543,471.08722])$

(%i4) first(eigenvalues(m)), ratprint : false$

(%i5) nfloat(% - conjugate(%),[],100);

(%o5) [8.0266455652163197256568351091[46 digits]5913348171925384517960952b-197*%i-1.3377742608693866209428058515[46 digits]0985558028654230752993492b-197,-5.3510970434775464837712234061[46 digits]3942232114616923011973968b-197*%i-1.3377742608693866209428058515[46 digits]0985558028654230752993492b-197,-1.9934389902195135071021405630[46 digits]0374693317196116973450023b-205*%i-2.6755485217387732418856117030[46 digits]1971116057308461505986984b-197]