  We report measurements of the isotope shifts of two transitions
($4f^46s\rightarrow [25044.7]^{\circ}_{7/2}$ and $4f^46s\rightarrow
[25138.6]^{\circ}_{7/2}$) in neodymium ions (Nd$^+$) with hundredfold improved
accuracy, using laser spectroscopy of a cryogenically-cooled neutral plasma.
The isotope shifts were measured across a set of five spin-zero isotopes that
spans a nuclear shape transition. We discuss the prospects for further
improvements to the accuracy of Nd$^+$ isotope shifts using optical clock
transitions, which could enable higher precision tests of King plot linearity
for new physics searches.
