Do you know specifically what the formula of cobalt chloride (eg. CoCl2) is that you're dealing with / asked?
i) Cobalt(III) could possibly oxidise alizarin, which would result in decolouring of this intensely red dye, and production of Co(II) [Co2+: Co3+(aq) + e- --> Co2+(aq) Co3+ is the oxidant and is itself reduced
or (my best bet):
ii) Cobalt(II or III?) chloride dissolved in water is providing Co2+ or Co3+ which may form a co-ordination complex with alizarin (the sodium part doesn't matter - its only the counter-ion there to balance charge).