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$\mathrm{Ti}^{2+}$ is purple while $\mathrm{Ti}^{4+}$ is colourless because
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$\mathrm{Ti}^{2+}$ has $3 d^2$ configuration
$\begin{aligned}
\mathrm{Ti}^{2+} & =[\mathrm{Ar}] 3 d^2 \\
\mathrm{Ti}^{4+} & =[\mathrm{Ar}] 3 d^0
\end{aligned}$
$\mathrm{Ti}^{2+}$ has two unpaired electrons in $3 d$ orbital and hence $d-d$ transition is possible due to absorption of light in visible region whereas $\mathrm{Ti}^{4+}$ is diamagnetic hence colourless.
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