Following up on the ideas in @Nick's answer and comments, you can see how wide the bandwidth needs to be to just flatten out the secondary mode:
Take this kernel density estimate as the proximal null—the distribution closest to the data yet still consistent with the null hypothesis that it's a sample from a unimodal population—and simulate from it. In the simulated samples the secondary mode doesn't often look so distinct, and you needn't widen the bandwidth as much to flatten it out.
Formalizing this approach leads to the test given in Silverman (1981), "Using kernel density estimates to investigate modality", JRSS B, 43, 1. Schwaiger & Holzmann's silvermantest package implements this test, and also the calibration procedure described by Hall & York (2001), "On the calibration of Silverman's test for multimodality", Statistica Sinica, 11, p 515, which adjusts for asymptotic conservatism. Performing the test on your data with a null hypothesis of unimodality results in p-values of 0.08 without calibration and 0.02 with calibration. I'm not familiar enough with the dip test to guess at why it might differ.
R code:
# kernel density estimate for x using Sheather-Jones method to estimate b/w:
density(x, kernel="gaussian", bw="SJ") -> dens.SJ
# tweak b/w until mode just disappears:
density(x, kernel="gaussian", bw=3160) -> prox.null
# fill matrix with simulated samples from the proximal null:
x.sim <- matrix(NA, nrow=length(x), ncol=10)
for (i in 1:10){
x.sim[ ,i] <- rnorm(length(x), sample(x, size=length(x), replace=T), prox.null$bw)
}
# perform Silverman test without Hall-York calibration:
require(silvermantest)
silverman.test(x, k=1, M=10000, adjust=F)
# perform Silverman test with Hall-York calibration:
silverman.test(x, k=1, M=10000, adjust=T)
