// Note the alternation
private static final Pattern PATTERN
    = Pattern.compile("\$[^.$]+(\.[^.$]+)*\$|[^.]+");

//

public List<String> matchesForInput(final String input)
{
    final Matcher m = PATTERN.matcher(input);
    final List<String> matches = new ArrayList<>();

    while (m.find())
        matches.add(m.group());

    return matches;
}

$[^$]+$|[[^]]+]|[^.]+

String s = "House.Car2.$4.45$.[0]";
Pattern pattern = Pattern.compile("\$[^$]+\$|\[[^\]]+\]|[^.]+");
Matcher matcher = pattern.matcher(s);
while (matcher.find()) {
   System.out.println(matcher.group());
}

House
Car2
$4.45$
[0]

public static List<String> parse(String input){
    //list which will hold retuned tokens
    List<String> tokens = new ArrayList<>();

    // flags representing if currently tested character is inside some of
    // special areas
    // (at start we are outside of these areas so hey are set to false)
    boolean insideDolar = false;          // $...$
    boolean insideSquareBrackets = false; // [...]
    boolean insideAgleBrackets =false;    // <...>

    // we need some buffer to build tokens, StringBuilder is excellent here
    StringBuilder sb = new StringBuilder();

    // now lets iterate over all characters and decide if we need to add them
    // to token or just add token to result list
    for (char ch : input.toCharArray()){

    // lets update in which area are we
        // finding $ means that we either start or end `$...$` area so
        // simple negation of flag is enough to update its status
        if (ch == '$') insideDolar = !insideDolar;
        //updating rest of flags seems pretty obvious
        else if (ch == '[') insideSquareBrackets = true;
        else if (ch == ']') insideSquareBrackets = false;
        else if (ch == '<') insideAgleBrackets = true;
        else if (ch == '>') insideAgleBrackets = false;

        // So now we know in which area we are, so lets handle special cases
        // if we are handling no dot
        // OR we are handling dot but we are inside either of areas we need
        // to just add it to token (append it to StringBuilder)
        if (ch != '.' || insideAgleBrackets|| insideDolar || insideSquareBrackets ){
            sb.append(ch);
        }else{// other case means that we are handling dot outside of special
              // areas where dots are not separators, so now they represents place
              // to split which means that we don't add it to token, but
              // add value from buffer (current token) to results and reset buffer
              // for next token
            tokens.add(sb.toString());
            sb.delete(0, sb.length());
        }
    }
    // also since we only add value held in buffer to list of tokens when we
    // find dot on which we split, there is high chance that we will not add
    // last token to result, because there is no dot after it, so we need to
    // do it manually after iterating over all characters
    if (sb.length()>0)//non empty token needs to be added to result
        tokens.add(sb.toString());

    return tokens;
}

String  input = "House.Car2.$abc.def$<ghi.jk>.[0]";
for (String s: parse(input))
    System.out.println(s);

House
Car2
$abc.def$<ghi.jk>
[0]