Hydrogen peroxide: catalyst changes colour during peroxide decomposition
Recently, I answered your question and relevant information can be found from that answer.

At first, I thought that the brown colour is due to ferrous chloride that formed during the course of reaction but since the reaction is occuring in a aqueous solution, it should be green in colour. Ferrous chloride is brown in anhydrous form. So I thought to be either due to formation of a complex, iron(II, III) oxide($\ce{Fe2O3}$), or iron(III) oxide-hydroxide($\ce{FeOOH.nH2O}$) or a mixed salt of oxide-chloride i.e. iron oxychloride($\ce{FeOCl}$).

The following information is from the paper 2 of that answer:


  According to the mechanism proposed by KREMER and STEIN, an
  intermediate oxygen complex of iron with oxidation number +V is
  primarily formed by the reaction of $\ce{Fe^3+}$ with $\ce{H2O2}$.
  This complex reacts with another $\ce{H2O2}$ molecule to water and
  oxygen thereby reforming $\ce{Fe^3+}$.
  
  $$\ce{Fe^3+ + H2O2 &lt;=&gt; [Fe^{III}OOH]^2+ + 2H+ &lt;=&gt; [Fe^{V}O]^3+ + H2O -&gt;[H2O2] Fe^3+ + 2H2O + O2}$$
  
  According to the mechanism proposed by HABER and WEISS the $\ce{Fe^3+}$ ions
  initiate a radical reaction, after which the chain reaction consumes
  the hydrogen peroxide. This mechanism can explain the high reaction
  rate very well.
  
  Chain initiation:  $\ce{Fe^3+ + H2O2 &lt;=&gt; [Fe^{III}OOH]^2+ + 2H+ &lt;=&gt; Fe^2+ + HOO. + H+}$
  
  Chain propagation: $\ce{Fe^2+ + H2O2 -&gt; Fe^3+ + 2OH.}$ 
  $\ce{Fe^3+ + H2O2 + OH. -&gt; Fe^3+ + HOO. + H2O -&gt; Fe^2+ + H+ + O2 + H2O }$


That intermediate oxygen complex of iron with oxidation number +V maybe responsible for the brown colour of solution.



As for the potassium iodide, the color you are seeing is basically food coloring. Various food coloring  is used to make the foam produced colorful. If food-color is not used, still color is seen. That color is basically color of pure soap foam. This elephant toothpaste experiment not uses food-color but still it has color that is color of soap-foam.