AnotherOne way to do this is to define the solution as
sol = Solve[b^2 + b*z + 1 == 0, b][[1]]; and then plot using the strategy suggested in the Help file for Solve, which replaces the variable in the rule (in this case b) with the desired solution:
Plot[b /. sol, {z, -10, 10}, PlotRange -> All]